| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | addsuniflem.3 | . . . 4
⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅)) | 
| 2 |  | addsuniflem.1 | . . . . 5
⊢ (𝜑 → 𝐿 <<s 𝑅) | 
| 3 | 2 | scutcld 27848 | . . . 4
⊢ (𝜑 → (𝐿 |s 𝑅) ∈  No
) | 
| 4 | 1, 3 | eqeltrd 2841 | . . 3
⊢ (𝜑 → 𝐴 ∈  No
) | 
| 5 |  | addsuniflem.4 | . . . 4
⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆)) | 
| 6 |  | addsuniflem.2 | . . . . 5
⊢ (𝜑 → 𝑀 <<s 𝑆) | 
| 7 | 6 | scutcld 27848 | . . . 4
⊢ (𝜑 → (𝑀 |s 𝑆) ∈  No
) | 
| 8 | 5, 7 | eqeltrd 2841 | . . 3
⊢ (𝜑 → 𝐵 ∈  No
) | 
| 9 |  | addsval2 27996 | . . 3
⊢ ((𝐴 ∈ 
No  ∧ 𝐵 ∈
 No ) → (𝐴 +s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))) | 
| 10 | 4, 8, 9 | syl2anc 584 | . 2
⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))) | 
| 11 | 4, 8 | addscut 28011 | . . . . 5
⊢ (𝜑 → ((𝐴 +s 𝐵) ∈  No 
∧ ({𝑎 ∣
∃𝑝 ∈ ( L
‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))) | 
| 12 | 11 | simp2d 1144 | . . . 4
⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s {(𝐴 +s 𝐵)}) | 
| 13 | 11 | simp3d 1145 | . . . 4
⊢ (𝜑 → {(𝐴 +s 𝐵)} <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) | 
| 14 |  | ovex 7464 | . . . . . 6
⊢ (𝐴 +s 𝐵) ∈ V | 
| 15 | 14 | snnz 4776 | . . . . 5
⊢ {(𝐴 +s 𝐵)} ≠ ∅ | 
| 16 |  | sslttr 27852 | . . . . 5
⊢ ((({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}) ∧ {(𝐴 +s 𝐵)} ≠ ∅) → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) | 
| 17 | 15, 16 | mp3an3 1452 | . . . 4
⊢ ((({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) | 
| 18 | 12, 13, 17 | syl2anc 584 | . . 3
⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) <<s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) | 
| 19 | 2, 1 | cofcutr1d 27959 | . . . . . 6
⊢ (𝜑 → ∀𝑝 ∈ ( L ‘𝐴)∃𝑙 ∈ 𝐿 𝑝 ≤s 𝑙) | 
| 20 |  | leftssno 27919 | . . . . . . . . . . 11
⊢ ( L
‘𝐴) ⊆  No | 
| 21 | 20 | sseli 3979 | . . . . . . . . . 10
⊢ (𝑝 ∈ ( L ‘𝐴) → 𝑝 ∈  No
) | 
| 22 | 21 | ad2antlr 727 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) ∧ 𝑙 ∈ 𝐿) → 𝑝 ∈  No
) | 
| 23 |  | ssltss1 27833 | . . . . . . . . . . . 12
⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆  No
) | 
| 24 | 2, 23 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐿 ⊆  No
) | 
| 25 | 24 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) → 𝐿 ⊆  No
) | 
| 26 | 25 | sselda 3983 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) ∧ 𝑙 ∈ 𝐿) → 𝑙 ∈  No
) | 
| 27 | 8 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) ∧ 𝑙 ∈ 𝐿) → 𝐵 ∈  No
) | 
| 28 | 22, 26, 27 | sleadd1d 28028 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) ∧ 𝑙 ∈ 𝐿) → (𝑝 ≤s 𝑙 ↔ (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵))) | 
| 29 | 28 | rexbidva 3177 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ( L ‘𝐴)) → (∃𝑙 ∈ 𝐿 𝑝 ≤s 𝑙 ↔ ∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵))) | 
| 30 | 29 | ralbidva 3176 | . . . . . 6
⊢ (𝜑 → (∀𝑝 ∈ ( L ‘𝐴)∃𝑙 ∈ 𝐿 𝑝 ≤s 𝑙 ↔ ∀𝑝 ∈ ( L ‘𝐴)∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵))) | 
| 31 | 19, 30 | mpbid 232 | . . . . 5
⊢ (𝜑 → ∀𝑝 ∈ ( L ‘𝐴)∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵)) | 
| 32 |  | eqeq1 2741 | . . . . . . . . . 10
⊢ (𝑦 = 𝑠 → (𝑦 = (𝑙 +s 𝐵) ↔ 𝑠 = (𝑙 +s 𝐵))) | 
| 33 | 32 | rexbidv 3179 | . . . . . . . . 9
⊢ (𝑦 = 𝑠 → (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) ↔ ∃𝑙 ∈ 𝐿 𝑠 = (𝑙 +s 𝐵))) | 
| 34 | 33 | rexab 3700 | . . . . . . . 8
⊢
(∃𝑠 ∈
{𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} (𝑝 +s 𝐵) ≤s 𝑠 ↔ ∃𝑠(∃𝑙 ∈ 𝐿 𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠)) | 
| 35 |  | rexcom4 3288 | . . . . . . . . 9
⊢
(∃𝑙 ∈
𝐿 ∃𝑠(𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ ∃𝑠∃𝑙 ∈ 𝐿 (𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠)) | 
| 36 |  | ovex 7464 | . . . . . . . . . . 11
⊢ (𝑙 +s 𝐵) ∈ V | 
| 37 |  | breq2 5147 | . . . . . . . . . . 11
⊢ (𝑠 = (𝑙 +s 𝐵) → ((𝑝 +s 𝐵) ≤s 𝑠 ↔ (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵))) | 
| 38 | 36, 37 | ceqsexv 3532 | . . . . . . . . . 10
⊢
(∃𝑠(𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵)) | 
| 39 | 38 | rexbii 3094 | . . . . . . . . 9
⊢
(∃𝑙 ∈
𝐿 ∃𝑠(𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ ∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵)) | 
| 40 |  | r19.41v 3189 | . . . . . . . . . 10
⊢
(∃𝑙 ∈
𝐿 (𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ (∃𝑙 ∈ 𝐿 𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠)) | 
| 41 | 40 | exbii 1848 | . . . . . . . . 9
⊢
(∃𝑠∃𝑙 ∈ 𝐿 (𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ ∃𝑠(∃𝑙 ∈ 𝐿 𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠)) | 
| 42 | 35, 39, 41 | 3bitr3ri 302 | . . . . . . . 8
⊢
(∃𝑠(∃𝑙 ∈ 𝐿 𝑠 = (𝑙 +s 𝐵) ∧ (𝑝 +s 𝐵) ≤s 𝑠) ↔ ∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵)) | 
| 43 | 34, 42 | bitri 275 | . . . . . . 7
⊢
(∃𝑠 ∈
{𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} (𝑝 +s 𝐵) ≤s 𝑠 ↔ ∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵)) | 
| 44 |  | ssun1 4178 | . . . . . . . 8
⊢ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ⊆ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) | 
| 45 |  | ssrexv 4053 | . . . . . . . 8
⊢ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ⊆ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) → (∃𝑠 ∈ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} (𝑝 +s 𝐵) ≤s 𝑠 → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠)) | 
| 46 | 44, 45 | ax-mp 5 | . . . . . . 7
⊢
(∃𝑠 ∈
{𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} (𝑝 +s 𝐵) ≤s 𝑠 → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) | 
| 47 | 43, 46 | sylbir 235 | . . . . . 6
⊢
(∃𝑙 ∈
𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) | 
| 48 | 47 | ralimi 3083 | . . . . 5
⊢
(∀𝑝 ∈ (
L ‘𝐴)∃𝑙 ∈ 𝐿 (𝑝 +s 𝐵) ≤s (𝑙 +s 𝐵) → ∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) | 
| 49 | 31, 48 | syl 17 | . . . 4
⊢ (𝜑 → ∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) | 
| 50 | 6, 5 | cofcutr1d 27959 | . . . . . 6
⊢ (𝜑 → ∀𝑞 ∈ ( L ‘𝐵)∃𝑚 ∈ 𝑀 𝑞 ≤s 𝑚) | 
| 51 |  | leftssno 27919 | . . . . . . . . . . 11
⊢ ( L
‘𝐵) ⊆  No | 
| 52 | 51 | sseli 3979 | . . . . . . . . . 10
⊢ (𝑞 ∈ ( L ‘𝐵) → 𝑞 ∈  No
) | 
| 53 | 52 | ad2antlr 727 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ 𝑚 ∈ 𝑀) → 𝑞 ∈  No
) | 
| 54 |  | ssltss1 27833 | . . . . . . . . . . . 12
⊢ (𝑀 <<s 𝑆 → 𝑀 ⊆  No
) | 
| 55 | 6, 54 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ⊆  No
) | 
| 56 | 55 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) → 𝑀 ⊆  No
) | 
| 57 | 56 | sselda 3983 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ 𝑚 ∈ 𝑀) → 𝑚 ∈  No
) | 
| 58 | 4 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ 𝑚 ∈ 𝑀) → 𝐴 ∈  No
) | 
| 59 | 53, 57, 58 | sleadd2d 28029 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ 𝑚 ∈ 𝑀) → (𝑞 ≤s 𝑚 ↔ (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚))) | 
| 60 | 59 | rexbidva 3177 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑞 ∈ ( L ‘𝐵)) → (∃𝑚 ∈ 𝑀 𝑞 ≤s 𝑚 ↔ ∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚))) | 
| 61 | 60 | ralbidva 3176 | . . . . . 6
⊢ (𝜑 → (∀𝑞 ∈ ( L ‘𝐵)∃𝑚 ∈ 𝑀 𝑞 ≤s 𝑚 ↔ ∀𝑞 ∈ ( L ‘𝐵)∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚))) | 
| 62 | 50, 61 | mpbid 232 | . . . . 5
⊢ (𝜑 → ∀𝑞 ∈ ( L ‘𝐵)∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚)) | 
| 63 |  | eqeq1 2741 | . . . . . . . . . 10
⊢ (𝑧 = 𝑠 → (𝑧 = (𝐴 +s 𝑚) ↔ 𝑠 = (𝐴 +s 𝑚))) | 
| 64 | 63 | rexbidv 3179 | . . . . . . . . 9
⊢ (𝑧 = 𝑠 → (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) ↔ ∃𝑚 ∈ 𝑀 𝑠 = (𝐴 +s 𝑚))) | 
| 65 | 64 | rexab 3700 | . . . . . . . 8
⊢
(∃𝑠 ∈
{𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} (𝐴 +s 𝑞) ≤s 𝑠 ↔ ∃𝑠(∃𝑚 ∈ 𝑀 𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠)) | 
| 66 |  | rexcom4 3288 | . . . . . . . . 9
⊢
(∃𝑚 ∈
𝑀 ∃𝑠(𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ ∃𝑠∃𝑚 ∈ 𝑀 (𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠)) | 
| 67 |  | ovex 7464 | . . . . . . . . . . 11
⊢ (𝐴 +s 𝑚) ∈ V | 
| 68 |  | breq2 5147 | . . . . . . . . . . 11
⊢ (𝑠 = (𝐴 +s 𝑚) → ((𝐴 +s 𝑞) ≤s 𝑠 ↔ (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚))) | 
| 69 | 67, 68 | ceqsexv 3532 | . . . . . . . . . 10
⊢
(∃𝑠(𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚)) | 
| 70 | 69 | rexbii 3094 | . . . . . . . . 9
⊢
(∃𝑚 ∈
𝑀 ∃𝑠(𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ ∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚)) | 
| 71 |  | r19.41v 3189 | . . . . . . . . . 10
⊢
(∃𝑚 ∈
𝑀 (𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ (∃𝑚 ∈ 𝑀 𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠)) | 
| 72 | 71 | exbii 1848 | . . . . . . . . 9
⊢
(∃𝑠∃𝑚 ∈ 𝑀 (𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ ∃𝑠(∃𝑚 ∈ 𝑀 𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠)) | 
| 73 | 66, 70, 72 | 3bitr3ri 302 | . . . . . . . 8
⊢
(∃𝑠(∃𝑚 ∈ 𝑀 𝑠 = (𝐴 +s 𝑚) ∧ (𝐴 +s 𝑞) ≤s 𝑠) ↔ ∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚)) | 
| 74 | 65, 73 | bitri 275 | . . . . . . 7
⊢
(∃𝑠 ∈
{𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} (𝐴 +s 𝑞) ≤s 𝑠 ↔ ∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚)) | 
| 75 |  | ssun2 4179 | . . . . . . . 8
⊢ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} ⊆ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) | 
| 76 |  | ssrexv 4053 | . . . . . . . 8
⊢ ({𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} ⊆ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) → (∃𝑠 ∈ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} (𝐴 +s 𝑞) ≤s 𝑠 → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠)) | 
| 77 | 75, 76 | ax-mp 5 | . . . . . . 7
⊢
(∃𝑠 ∈
{𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} (𝐴 +s 𝑞) ≤s 𝑠 → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) | 
| 78 | 74, 77 | sylbir 235 | . . . . . 6
⊢
(∃𝑚 ∈
𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) | 
| 79 | 78 | ralimi 3083 | . . . . 5
⊢
(∀𝑞 ∈ (
L ‘𝐵)∃𝑚 ∈ 𝑀 (𝐴 +s 𝑞) ≤s (𝐴 +s 𝑚) → ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) | 
| 80 | 62, 79 | syl 17 | . . . 4
⊢ (𝜑 → ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) | 
| 81 |  | ralunb 4197 | . . . . 5
⊢
(∀𝑟 ∈
({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)})∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ (∀𝑟 ∈ {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ∧ ∀𝑟 ∈ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) | 
| 82 |  | eqeq1 2741 | . . . . . . . . 9
⊢ (𝑎 = 𝑟 → (𝑎 = (𝑝 +s 𝐵) ↔ 𝑟 = (𝑝 +s 𝐵))) | 
| 83 | 82 | rexbidv 3179 | . . . . . . . 8
⊢ (𝑎 = 𝑟 → (∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵) ↔ ∃𝑝 ∈ ( L ‘𝐴)𝑟 = (𝑝 +s 𝐵))) | 
| 84 | 83 | ralab 3697 | . . . . . . 7
⊢
(∀𝑟 ∈
{𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∀𝑟(∃𝑝 ∈ ( L ‘𝐴)𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) | 
| 85 |  | ralcom4 3286 | . . . . . . . 8
⊢
(∀𝑝 ∈ (
L ‘𝐴)∀𝑟(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑟∀𝑝 ∈ ( L ‘𝐴)(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) | 
| 86 |  | ovex 7464 | . . . . . . . . . 10
⊢ (𝑝 +s 𝐵) ∈ V | 
| 87 |  | breq1 5146 | . . . . . . . . . . 11
⊢ (𝑟 = (𝑝 +s 𝐵) → (𝑟 ≤s 𝑠 ↔ (𝑝 +s 𝐵) ≤s 𝑠)) | 
| 88 | 87 | rexbidv 3179 | . . . . . . . . . 10
⊢ (𝑟 = (𝑝 +s 𝐵) → (∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠)) | 
| 89 | 86, 88 | ceqsalv 3521 | . . . . . . . . 9
⊢
(∀𝑟(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) | 
| 90 | 89 | ralbii 3093 | . . . . . . . 8
⊢
(∀𝑝 ∈ (
L ‘𝐴)∀𝑟(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) | 
| 91 |  | r19.23v 3183 | . . . . . . . . 9
⊢
(∀𝑝 ∈ (
L ‘𝐴)(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ (∃𝑝 ∈ ( L ‘𝐴)𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) | 
| 92 | 91 | albii 1819 | . . . . . . . 8
⊢
(∀𝑟∀𝑝 ∈ ( L ‘𝐴)(𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑟(∃𝑝 ∈ ( L ‘𝐴)𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) | 
| 93 | 85, 90, 92 | 3bitr3ri 302 | . . . . . . 7
⊢
(∀𝑟(∃𝑝 ∈ ( L ‘𝐴)𝑟 = (𝑝 +s 𝐵) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) | 
| 94 | 84, 93 | bitri 275 | . . . . . 6
⊢
(∀𝑟 ∈
{𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠) | 
| 95 |  | eqeq1 2741 | . . . . . . . . 9
⊢ (𝑏 = 𝑟 → (𝑏 = (𝐴 +s 𝑞) ↔ 𝑟 = (𝐴 +s 𝑞))) | 
| 96 | 95 | rexbidv 3179 | . . . . . . . 8
⊢ (𝑏 = 𝑟 → (∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞) ↔ ∃𝑞 ∈ ( L ‘𝐵)𝑟 = (𝐴 +s 𝑞))) | 
| 97 | 96 | ralab 3697 | . . . . . . 7
⊢
(∀𝑟 ∈
{𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∀𝑟(∃𝑞 ∈ ( L ‘𝐵)𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) | 
| 98 |  | ralcom4 3286 | . . . . . . . 8
⊢
(∀𝑞 ∈ (
L ‘𝐵)∀𝑟(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑟∀𝑞 ∈ ( L ‘𝐵)(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) | 
| 99 |  | ovex 7464 | . . . . . . . . . 10
⊢ (𝐴 +s 𝑞) ∈ V | 
| 100 |  | breq1 5146 | . . . . . . . . . . 11
⊢ (𝑟 = (𝐴 +s 𝑞) → (𝑟 ≤s 𝑠 ↔ (𝐴 +s 𝑞) ≤s 𝑠)) | 
| 101 | 100 | rexbidv 3179 | . . . . . . . . . 10
⊢ (𝑟 = (𝐴 +s 𝑞) → (∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠)) | 
| 102 | 99, 101 | ceqsalv 3521 | . . . . . . . . 9
⊢
(∀𝑟(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) | 
| 103 | 102 | ralbii 3093 | . . . . . . . 8
⊢
(∀𝑞 ∈ (
L ‘𝐵)∀𝑟(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) | 
| 104 |  | r19.23v 3183 | . . . . . . . . 9
⊢
(∀𝑞 ∈ (
L ‘𝐵)(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ (∃𝑞 ∈ ( L ‘𝐵)𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) | 
| 105 | 104 | albii 1819 | . . . . . . . 8
⊢
(∀𝑟∀𝑞 ∈ ( L ‘𝐵)(𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑟(∃𝑞 ∈ ( L ‘𝐵)𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠)) | 
| 106 | 98, 103, 105 | 3bitr3ri 302 | . . . . . . 7
⊢
(∀𝑟(∃𝑞 ∈ ( L ‘𝐵)𝑟 = (𝐴 +s 𝑞) → ∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) | 
| 107 | 97, 106 | bitri 275 | . . . . . 6
⊢
(∀𝑟 ∈
{𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠) | 
| 108 | 94, 107 | anbi12i 628 | . . . . 5
⊢
((∀𝑟 ∈
{𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ∧ ∀𝑟 ∈ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) ↔ (∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠 ∧ ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠)) | 
| 109 | 81, 108 | bitri 275 | . . . 4
⊢
(∀𝑟 ∈
({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)})∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠 ↔ (∀𝑝 ∈ ( L ‘𝐴)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝑝 +s 𝐵) ≤s 𝑠 ∧ ∀𝑞 ∈ ( L ‘𝐵)∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})(𝐴 +s 𝑞) ≤s 𝑠)) | 
| 110 | 49, 80, 109 | sylanbrc 583 | . . 3
⊢ (𝜑 → ∀𝑟 ∈ ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)})∃𝑠 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})𝑟 ≤s 𝑠) | 
| 111 | 2, 1 | cofcutr2d 27960 | . . . . . 6
⊢ (𝜑 → ∀𝑒 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑒) | 
| 112 |  | ssltss2 27834 | . . . . . . . . . . . 12
⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆  No
) | 
| 113 | 2, 112 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ⊆  No
) | 
| 114 | 113 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) → 𝑅 ⊆  No
) | 
| 115 | 114 | sselda 3983 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈  No
) | 
| 116 |  | rightssno 27920 | . . . . . . . . . . 11
⊢ ( R
‘𝐴) ⊆  No | 
| 117 | 116 | sseli 3979 | . . . . . . . . . 10
⊢ (𝑒 ∈ ( R ‘𝐴) → 𝑒 ∈  No
) | 
| 118 | 117 | ad2antlr 727 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) ∧ 𝑟 ∈ 𝑅) → 𝑒 ∈  No
) | 
| 119 | 8 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) ∧ 𝑟 ∈ 𝑅) → 𝐵 ∈  No
) | 
| 120 | 115, 118,
119 | sleadd1d 28028 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ≤s 𝑒 ↔ (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵))) | 
| 121 | 120 | rexbidva 3177 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑒 ∈ ( R ‘𝐴)) → (∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑒 ↔ ∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵))) | 
| 122 | 121 | ralbidva 3176 | . . . . . 6
⊢ (𝜑 → (∀𝑒 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑒 ↔ ∀𝑒 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵))) | 
| 123 | 111, 122 | mpbid 232 | . . . . 5
⊢ (𝜑 → ∀𝑒 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵)) | 
| 124 |  | eqeq1 2741 | . . . . . . . . . 10
⊢ (𝑤 = 𝑏 → (𝑤 = (𝑟 +s 𝐵) ↔ 𝑏 = (𝑟 +s 𝐵))) | 
| 125 | 124 | rexbidv 3179 | . . . . . . . . 9
⊢ (𝑤 = 𝑏 → (∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵) ↔ ∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵))) | 
| 126 | 125 | rexab 3700 | . . . . . . . 8
⊢
(∃𝑏 ∈
{𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)}𝑏 ≤s (𝑒 +s 𝐵) ↔ ∃𝑏(∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵))) | 
| 127 |  | rexcom4 3288 | . . . . . . . . 9
⊢
(∃𝑟 ∈
𝑅 ∃𝑏(𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ ∃𝑏∃𝑟 ∈ 𝑅 (𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵))) | 
| 128 |  | ovex 7464 | . . . . . . . . . . 11
⊢ (𝑟 +s 𝐵) ∈ V | 
| 129 |  | breq1 5146 | . . . . . . . . . . 11
⊢ (𝑏 = (𝑟 +s 𝐵) → (𝑏 ≤s (𝑒 +s 𝐵) ↔ (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵))) | 
| 130 | 128, 129 | ceqsexv 3532 | . . . . . . . . . 10
⊢
(∃𝑏(𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵)) | 
| 131 | 130 | rexbii 3094 | . . . . . . . . 9
⊢
(∃𝑟 ∈
𝑅 ∃𝑏(𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ ∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵)) | 
| 132 |  | r19.41v 3189 | . . . . . . . . . 10
⊢
(∃𝑟 ∈
𝑅 (𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ (∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵))) | 
| 133 | 132 | exbii 1848 | . . . . . . . . 9
⊢
(∃𝑏∃𝑟 ∈ 𝑅 (𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ ∃𝑏(∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵))) | 
| 134 | 127, 131,
133 | 3bitr3ri 302 | . . . . . . . 8
⊢
(∃𝑏(∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∧ 𝑏 ≤s (𝑒 +s 𝐵)) ↔ ∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵)) | 
| 135 | 126, 134 | bitri 275 | . . . . . . 7
⊢
(∃𝑏 ∈
{𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)}𝑏 ≤s (𝑒 +s 𝐵) ↔ ∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵)) | 
| 136 |  | ssun1 4178 | . . . . . . . 8
⊢ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ⊆ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) | 
| 137 |  | ssrexv 4053 | . . . . . . . 8
⊢ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ⊆ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → (∃𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)}𝑏 ≤s (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵))) | 
| 138 | 136, 137 | ax-mp 5 | . . . . . . 7
⊢
(∃𝑏 ∈
{𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)}𝑏 ≤s (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) | 
| 139 | 135, 138 | sylbir 235 | . . . . . 6
⊢
(∃𝑟 ∈
𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) | 
| 140 | 139 | ralimi 3083 | . . . . 5
⊢
(∀𝑒 ∈ (
R ‘𝐴)∃𝑟 ∈ 𝑅 (𝑟 +s 𝐵) ≤s (𝑒 +s 𝐵) → ∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) | 
| 141 | 123, 140 | syl 17 | . . . 4
⊢ (𝜑 → ∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) | 
| 142 | 6, 5 | cofcutr2d 27960 | . . . . . 6
⊢ (𝜑 → ∀𝑓 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑓) | 
| 143 |  | ssltss2 27834 | . . . . . . . . . . . 12
⊢ (𝑀 <<s 𝑆 → 𝑆 ⊆  No
) | 
| 144 | 6, 143 | syl 17 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑆 ⊆  No
) | 
| 145 | 144 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) → 𝑆 ⊆  No
) | 
| 146 | 145 | sselda 3983 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈  No
) | 
| 147 |  | rightssno 27920 | . . . . . . . . . . 11
⊢ ( R
‘𝐵) ⊆  No | 
| 148 | 147 | sseli 3979 | . . . . . . . . . 10
⊢ (𝑓 ∈ ( R ‘𝐵) → 𝑓 ∈  No
) | 
| 149 | 148 | ad2antlr 727 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) ∧ 𝑠 ∈ 𝑆) → 𝑓 ∈  No
) | 
| 150 | 4 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) ∧ 𝑠 ∈ 𝑆) → 𝐴 ∈  No
) | 
| 151 | 146, 149,
150 | sleadd2d 28029 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) ∧ 𝑠 ∈ 𝑆) → (𝑠 ≤s 𝑓 ↔ (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓))) | 
| 152 | 151 | rexbidva 3177 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ ( R ‘𝐵)) → (∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑓 ↔ ∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓))) | 
| 153 | 152 | ralbidva 3176 | . . . . . 6
⊢ (𝜑 → (∀𝑓 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑓 ↔ ∀𝑓 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓))) | 
| 154 | 142, 153 | mpbid 232 | . . . . 5
⊢ (𝜑 → ∀𝑓 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓)) | 
| 155 |  | eqeq1 2741 | . . . . . . . . . 10
⊢ (𝑡 = 𝑏 → (𝑡 = (𝐴 +s 𝑠) ↔ 𝑏 = (𝐴 +s 𝑠))) | 
| 156 | 155 | rexbidv 3179 | . . . . . . . . 9
⊢ (𝑡 = 𝑏 → (∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠) ↔ ∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠))) | 
| 157 | 156 | rexab 3700 | . . . . . . . 8
⊢
(∃𝑏 ∈
{𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}𝑏 ≤s (𝐴 +s 𝑓) ↔ ∃𝑏(∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓))) | 
| 158 |  | rexcom4 3288 | . . . . . . . . 9
⊢
(∃𝑠 ∈
𝑆 ∃𝑏(𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ ∃𝑏∃𝑠 ∈ 𝑆 (𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓))) | 
| 159 |  | ovex 7464 | . . . . . . . . . . 11
⊢ (𝐴 +s 𝑠) ∈ V | 
| 160 |  | breq1 5146 | . . . . . . . . . . 11
⊢ (𝑏 = (𝐴 +s 𝑠) → (𝑏 ≤s (𝐴 +s 𝑓) ↔ (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓))) | 
| 161 | 159, 160 | ceqsexv 3532 | . . . . . . . . . 10
⊢
(∃𝑏(𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓)) | 
| 162 | 161 | rexbii 3094 | . . . . . . . . 9
⊢
(∃𝑠 ∈
𝑆 ∃𝑏(𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ ∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓)) | 
| 163 |  | r19.41v 3189 | . . . . . . . . . 10
⊢
(∃𝑠 ∈
𝑆 (𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ (∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓))) | 
| 164 | 163 | exbii 1848 | . . . . . . . . 9
⊢
(∃𝑏∃𝑠 ∈ 𝑆 (𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ ∃𝑏(∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓))) | 
| 165 | 158, 162,
164 | 3bitr3ri 302 | . . . . . . . 8
⊢
(∃𝑏(∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠) ∧ 𝑏 ≤s (𝐴 +s 𝑓)) ↔ ∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓)) | 
| 166 | 157, 165 | bitri 275 | . . . . . . 7
⊢
(∃𝑏 ∈
{𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}𝑏 ≤s (𝐴 +s 𝑓) ↔ ∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓)) | 
| 167 |  | ssun2 4179 | . . . . . . . 8
⊢ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} ⊆ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) | 
| 168 |  | ssrexv 4053 | . . . . . . . 8
⊢ ({𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} ⊆ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → (∃𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}𝑏 ≤s (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓))) | 
| 169 | 167, 168 | ax-mp 5 | . . . . . . 7
⊢
(∃𝑏 ∈
{𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}𝑏 ≤s (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) | 
| 170 | 166, 169 | sylbir 235 | . . . . . 6
⊢
(∃𝑠 ∈
𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) | 
| 171 | 170 | ralimi 3083 | . . . . 5
⊢
(∀𝑓 ∈ (
R ‘𝐵)∃𝑠 ∈ 𝑆 (𝐴 +s 𝑠) ≤s (𝐴 +s 𝑓) → ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) | 
| 172 | 154, 171 | syl 17 | . . . 4
⊢ (𝜑 → ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) | 
| 173 |  | ralunb 4197 | . . . . 5
⊢
(∀𝑎 ∈
({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ (∀𝑎 ∈ {𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ∧ ∀𝑎 ∈ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) | 
| 174 |  | eqeq1 2741 | . . . . . . . . 9
⊢ (𝑐 = 𝑎 → (𝑐 = (𝑒 +s 𝐵) ↔ 𝑎 = (𝑒 +s 𝐵))) | 
| 175 | 174 | rexbidv 3179 | . . . . . . . 8
⊢ (𝑐 = 𝑎 → (∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵) ↔ ∃𝑒 ∈ ( R ‘𝐴)𝑎 = (𝑒 +s 𝐵))) | 
| 176 | 175 | ralab 3697 | . . . . . . 7
⊢
(∀𝑎 ∈
{𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∀𝑎(∃𝑒 ∈ ( R ‘𝐴)𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) | 
| 177 |  | ralcom4 3286 | . . . . . . . 8
⊢
(∀𝑒 ∈ (
R ‘𝐴)∀𝑎(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑎∀𝑒 ∈ ( R ‘𝐴)(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) | 
| 178 |  | ovex 7464 | . . . . . . . . . 10
⊢ (𝑒 +s 𝐵) ∈ V | 
| 179 |  | breq2 5147 | . . . . . . . . . . 11
⊢ (𝑎 = (𝑒 +s 𝐵) → (𝑏 ≤s 𝑎 ↔ 𝑏 ≤s (𝑒 +s 𝐵))) | 
| 180 | 179 | rexbidv 3179 | . . . . . . . . . 10
⊢ (𝑎 = (𝑒 +s 𝐵) → (∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵))) | 
| 181 | 178, 180 | ceqsalv 3521 | . . . . . . . . 9
⊢
(∀𝑎(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) | 
| 182 | 181 | ralbii 3093 | . . . . . . . 8
⊢
(∀𝑒 ∈ (
R ‘𝐴)∀𝑎(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) | 
| 183 |  | r19.23v 3183 | . . . . . . . . 9
⊢
(∀𝑒 ∈ (
R ‘𝐴)(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ (∃𝑒 ∈ ( R ‘𝐴)𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) | 
| 184 | 183 | albii 1819 | . . . . . . . 8
⊢
(∀𝑎∀𝑒 ∈ ( R ‘𝐴)(𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑎(∃𝑒 ∈ ( R ‘𝐴)𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) | 
| 185 | 177, 182,
184 | 3bitr3ri 302 | . . . . . . 7
⊢
(∀𝑎(∃𝑒 ∈ ( R ‘𝐴)𝑎 = (𝑒 +s 𝐵) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) | 
| 186 | 176, 185 | bitri 275 | . . . . . 6
⊢
(∀𝑎 ∈
{𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵)) | 
| 187 |  | eqeq1 2741 | . . . . . . . . 9
⊢ (𝑑 = 𝑎 → (𝑑 = (𝐴 +s 𝑓) ↔ 𝑎 = (𝐴 +s 𝑓))) | 
| 188 | 187 | rexbidv 3179 | . . . . . . . 8
⊢ (𝑑 = 𝑎 → (∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓) ↔ ∃𝑓 ∈ ( R ‘𝐵)𝑎 = (𝐴 +s 𝑓))) | 
| 189 | 188 | ralab 3697 | . . . . . . 7
⊢
(∀𝑎 ∈
{𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∀𝑎(∃𝑓 ∈ ( R ‘𝐵)𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) | 
| 190 |  | ralcom4 3286 | . . . . . . . 8
⊢
(∀𝑓 ∈ (
R ‘𝐵)∀𝑎(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑎∀𝑓 ∈ ( R ‘𝐵)(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) | 
| 191 |  | ovex 7464 | . . . . . . . . . 10
⊢ (𝐴 +s 𝑓) ∈ V | 
| 192 |  | breq2 5147 | . . . . . . . . . . 11
⊢ (𝑎 = (𝐴 +s 𝑓) → (𝑏 ≤s 𝑎 ↔ 𝑏 ≤s (𝐴 +s 𝑓))) | 
| 193 | 192 | rexbidv 3179 | . . . . . . . . . 10
⊢ (𝑎 = (𝐴 +s 𝑓) → (∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓))) | 
| 194 | 191, 193 | ceqsalv 3521 | . . . . . . . . 9
⊢
(∀𝑎(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) | 
| 195 | 194 | ralbii 3093 | . . . . . . . 8
⊢
(∀𝑓 ∈ (
R ‘𝐵)∀𝑎(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) | 
| 196 |  | r19.23v 3183 | . . . . . . . . 9
⊢
(∀𝑓 ∈ (
R ‘𝐵)(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ (∃𝑓 ∈ ( R ‘𝐵)𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) | 
| 197 | 196 | albii 1819 | . . . . . . . 8
⊢
(∀𝑎∀𝑓 ∈ ( R ‘𝐵)(𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑎(∃𝑓 ∈ ( R ‘𝐵)𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎)) | 
| 198 | 190, 195,
197 | 3bitr3ri 302 | . . . . . . 7
⊢
(∀𝑎(∃𝑓 ∈ ( R ‘𝐵)𝑎 = (𝐴 +s 𝑓) → ∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) | 
| 199 | 189, 198 | bitri 275 | . . . . . 6
⊢
(∀𝑎 ∈
{𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓)) | 
| 200 | 186, 199 | anbi12i 628 | . . . . 5
⊢
((∀𝑎 ∈
{𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ∧ ∀𝑎 ∈ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) ↔ (∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵) ∧ ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓))) | 
| 201 | 173, 200 | bitri 275 | . . . 4
⊢
(∀𝑎 ∈
({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎 ↔ (∀𝑒 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝑒 +s 𝐵) ∧ ∀𝑓 ∈ ( R ‘𝐵)∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s (𝐴 +s 𝑓))) | 
| 202 | 141, 172,
201 | sylanbrc 583 | . . 3
⊢ (𝜑 → ∀𝑎 ∈ ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})∃𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})𝑏 ≤s 𝑎) | 
| 203 |  | eqid 2737 | . . . . . . . 8
⊢ (𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) = (𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) | 
| 204 | 203 | rnmpt 5968 | . . . . . . 7
⊢ ran
(𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) = {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} | 
| 205 |  | ssltex1 27831 | . . . . . . . . . 10
⊢ (𝐿 <<s 𝑅 → 𝐿 ∈ V) | 
| 206 | 2, 205 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝐿 ∈ V) | 
| 207 | 206 | mptexd 7244 | . . . . . . . 8
⊢ (𝜑 → (𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) ∈ V) | 
| 208 |  | rnexg 7924 | . . . . . . . 8
⊢ ((𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) ∈ V → ran (𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) ∈ V) | 
| 209 | 207, 208 | syl 17 | . . . . . . 7
⊢ (𝜑 → ran (𝑙 ∈ 𝐿 ↦ (𝑙 +s 𝐵)) ∈ V) | 
| 210 | 204, 209 | eqeltrrid 2846 | . . . . . 6
⊢ (𝜑 → {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∈ V) | 
| 211 |  | eqid 2737 | . . . . . . . 8
⊢ (𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) = (𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) | 
| 212 | 211 | rnmpt 5968 | . . . . . . 7
⊢ ran
(𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) = {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} | 
| 213 |  | ssltex1 27831 | . . . . . . . . . 10
⊢ (𝑀 <<s 𝑆 → 𝑀 ∈ V) | 
| 214 | 6, 213 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ V) | 
| 215 | 214 | mptexd 7244 | . . . . . . . 8
⊢ (𝜑 → (𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) ∈ V) | 
| 216 |  | rnexg 7924 | . . . . . . . 8
⊢ ((𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) ∈ V → ran (𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) ∈ V) | 
| 217 | 215, 216 | syl 17 | . . . . . . 7
⊢ (𝜑 → ran (𝑚 ∈ 𝑀 ↦ (𝐴 +s 𝑚)) ∈ V) | 
| 218 | 212, 217 | eqeltrrid 2846 | . . . . . 6
⊢ (𝜑 → {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} ∈ V) | 
| 219 | 210, 218 | unexd 7774 | . . . . 5
⊢ (𝜑 → ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ∈ V) | 
| 220 |  | snex 5436 | . . . . . 6
⊢ {(𝐴 +s 𝐵)} ∈ V | 
| 221 | 220 | a1i 11 | . . . . 5
⊢ (𝜑 → {(𝐴 +s 𝐵)} ∈ V) | 
| 222 | 24 | sselda 3983 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝑙 ∈  No
) | 
| 223 | 8 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝐵 ∈  No
) | 
| 224 | 222, 223 | addscld 28013 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝑙 +s 𝐵) ∈  No
) | 
| 225 |  | eleq1 2829 | . . . . . . . . 9
⊢ (𝑦 = (𝑙 +s 𝐵) → (𝑦 ∈  No 
↔ (𝑙 +s
𝐵) ∈  No )) | 
| 226 | 224, 225 | syl5ibrcom 247 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝑦 = (𝑙 +s 𝐵) → 𝑦 ∈  No
)) | 
| 227 | 226 | rexlimdva 3155 | . . . . . . 7
⊢ (𝜑 → (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) → 𝑦 ∈  No
)) | 
| 228 | 227 | abssdv 4068 | . . . . . 6
⊢ (𝜑 → {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ⊆  No
) | 
| 229 | 4 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝐴 ∈  No
) | 
| 230 | 55 | sselda 3983 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝑚 ∈  No
) | 
| 231 | 229, 230 | addscld 28013 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝐴 +s 𝑚) ∈  No
) | 
| 232 |  | eleq1 2829 | . . . . . . . . 9
⊢ (𝑧 = (𝐴 +s 𝑚) → (𝑧 ∈  No 
↔ (𝐴 +s
𝑚) ∈  No )) | 
| 233 | 231, 232 | syl5ibrcom 247 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝑧 = (𝐴 +s 𝑚) → 𝑧 ∈  No
)) | 
| 234 | 233 | rexlimdva 3155 | . . . . . . 7
⊢ (𝜑 → (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) → 𝑧 ∈  No
)) | 
| 235 | 234 | abssdv 4068 | . . . . . 6
⊢ (𝜑 → {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} ⊆  No
) | 
| 236 | 228, 235 | unssd 4192 | . . . . 5
⊢ (𝜑 → ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ⊆  No
) | 
| 237 | 4, 8 | addscld 28013 | . . . . . 6
⊢ (𝜑 → (𝐴 +s 𝐵) ∈  No
) | 
| 238 | 237 | snssd 4809 | . . . . 5
⊢ (𝜑 → {(𝐴 +s 𝐵)} ⊆  No
) | 
| 239 |  | velsn 4642 | . . . . . . 7
⊢ (𝑏 ∈ {(𝐴 +s 𝐵)} ↔ 𝑏 = (𝐴 +s 𝐵)) | 
| 240 |  | elun 4153 | . . . . . . . . . . 11
⊢ (𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ↔ (𝑎 ∈ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∨ 𝑎 ∈ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})) | 
| 241 |  | vex 3484 | . . . . . . . . . . . . 13
⊢ 𝑎 ∈ V | 
| 242 |  | eqeq1 2741 | . . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑎 → (𝑦 = (𝑙 +s 𝐵) ↔ 𝑎 = (𝑙 +s 𝐵))) | 
| 243 | 242 | rexbidv 3179 | . . . . . . . . . . . . 13
⊢ (𝑦 = 𝑎 → (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) ↔ ∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵))) | 
| 244 | 241, 243 | elab 3679 | . . . . . . . . . . . 12
⊢ (𝑎 ∈ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ↔ ∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵)) | 
| 245 |  | eqeq1 2741 | . . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑎 → (𝑧 = (𝐴 +s 𝑚) ↔ 𝑎 = (𝐴 +s 𝑚))) | 
| 246 | 245 | rexbidv 3179 | . . . . . . . . . . . . 13
⊢ (𝑧 = 𝑎 → (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) ↔ ∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚))) | 
| 247 | 241, 246 | elab 3679 | . . . . . . . . . . . 12
⊢ (𝑎 ∈ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} ↔ ∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚)) | 
| 248 | 244, 247 | orbi12i 915 | . . . . . . . . . . 11
⊢ ((𝑎 ∈ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∨ 𝑎 ∈ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ↔ (∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵) ∨ ∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚))) | 
| 249 | 240, 248 | bitri 275 | . . . . . . . . . 10
⊢ (𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ↔ (∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵) ∨ ∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚))) | 
| 250 |  | scutcut 27846 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈  No 
∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅)) | 
| 251 | 2, 250 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝐿 |s 𝑅) ∈  No 
∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅)) | 
| 252 | 251 | simp2d 1144 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐿 <<s {(𝐿 |s 𝑅)}) | 
| 253 | 252 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝐿 <<s {(𝐿 |s 𝑅)}) | 
| 254 |  | simpr 484 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝑙 ∈ 𝐿) | 
| 255 |  | ovex 7464 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐿 |s 𝑅) ∈ V | 
| 256 | 255 | snid 4662 | . . . . . . . . . . . . . . . . 17
⊢ (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)} | 
| 257 | 256 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)}) | 
| 258 | 253, 254,
257 | ssltsepcd 27839 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝑙 <s (𝐿 |s 𝑅)) | 
| 259 | 1 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝐴 = (𝐿 |s 𝑅)) | 
| 260 | 258, 259 | breqtrrd 5171 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝑙 <s 𝐴) | 
| 261 | 4 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → 𝐴 ∈  No
) | 
| 262 | 222, 261,
223 | sltadd1d 28031 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝑙 <s 𝐴 ↔ (𝑙 +s 𝐵) <s (𝐴 +s 𝐵))) | 
| 263 | 260, 262 | mpbid 232 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝑙 +s 𝐵) <s (𝐴 +s 𝐵)) | 
| 264 |  | breq1 5146 | . . . . . . . . . . . . 13
⊢ (𝑎 = (𝑙 +s 𝐵) → (𝑎 <s (𝐴 +s 𝐵) ↔ (𝑙 +s 𝐵) <s (𝐴 +s 𝐵))) | 
| 265 | 263, 264 | syl5ibrcom 247 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑙 ∈ 𝐿) → (𝑎 = (𝑙 +s 𝐵) → 𝑎 <s (𝐴 +s 𝐵))) | 
| 266 | 265 | rexlimdva 3155 | . . . . . . . . . . 11
⊢ (𝜑 → (∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵) → 𝑎 <s (𝐴 +s 𝐵))) | 
| 267 |  | scutcut 27846 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑀 <<s 𝑆 → ((𝑀 |s 𝑆) ∈  No 
∧ 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆)) | 
| 268 | 6, 267 | syl 17 | . . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑀 |s 𝑆) ∈  No 
∧ 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆)) | 
| 269 | 268 | simp2d 1144 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 <<s {(𝑀 |s 𝑆)}) | 
| 270 | 269 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝑀 <<s {(𝑀 |s 𝑆)}) | 
| 271 |  | simpr 484 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝑚 ∈ 𝑀) | 
| 272 |  | ovex 7464 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑀 |s 𝑆) ∈ V | 
| 273 | 272 | snid 4662 | . . . . . . . . . . . . . . . . 17
⊢ (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)} | 
| 274 | 273 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)}) | 
| 275 | 270, 271,
274 | ssltsepcd 27839 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝑚 <s (𝑀 |s 𝑆)) | 
| 276 | 5 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝐵 = (𝑀 |s 𝑆)) | 
| 277 | 275, 276 | breqtrrd 5171 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝑚 <s 𝐵) | 
| 278 | 8 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → 𝐵 ∈  No
) | 
| 279 | 230, 278,
229 | sltadd2d 28030 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝑚 <s 𝐵 ↔ (𝐴 +s 𝑚) <s (𝐴 +s 𝐵))) | 
| 280 | 277, 279 | mpbid 232 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝐴 +s 𝑚) <s (𝐴 +s 𝐵)) | 
| 281 |  | breq1 5146 | . . . . . . . . . . . . 13
⊢ (𝑎 = (𝐴 +s 𝑚) → (𝑎 <s (𝐴 +s 𝐵) ↔ (𝐴 +s 𝑚) <s (𝐴 +s 𝐵))) | 
| 282 | 280, 281 | syl5ibrcom 247 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ 𝑀) → (𝑎 = (𝐴 +s 𝑚) → 𝑎 <s (𝐴 +s 𝐵))) | 
| 283 | 282 | rexlimdva 3155 | . . . . . . . . . . 11
⊢ (𝜑 → (∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚) → 𝑎 <s (𝐴 +s 𝐵))) | 
| 284 | 266, 283 | jaod 860 | . . . . . . . . . 10
⊢ (𝜑 → ((∃𝑙 ∈ 𝐿 𝑎 = (𝑙 +s 𝐵) ∨ ∃𝑚 ∈ 𝑀 𝑎 = (𝐴 +s 𝑚)) → 𝑎 <s (𝐴 +s 𝐵))) | 
| 285 | 249, 284 | biimtrid 242 | . . . . . . . . 9
⊢ (𝜑 → (𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) → 𝑎 <s (𝐴 +s 𝐵))) | 
| 286 | 285 | imp 406 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})) → 𝑎 <s (𝐴 +s 𝐵)) | 
| 287 |  | breq2 5147 | . . . . . . . 8
⊢ (𝑏 = (𝐴 +s 𝐵) → (𝑎 <s 𝑏 ↔ 𝑎 <s (𝐴 +s 𝐵))) | 
| 288 | 286, 287 | syl5ibrcom 247 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})) → (𝑏 = (𝐴 +s 𝐵) → 𝑎 <s 𝑏)) | 
| 289 | 239, 288 | biimtrid 242 | . . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)})) → (𝑏 ∈ {(𝐴 +s 𝐵)} → 𝑎 <s 𝑏)) | 
| 290 | 289 | 3impia 1118 | . . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) ∧ 𝑏 ∈ {(𝐴 +s 𝐵)}) → 𝑎 <s 𝑏) | 
| 291 | 219, 221,
236, 238, 290 | ssltd 27836 | . . . 4
⊢ (𝜑 → ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)}) | 
| 292 | 10 | sneqd 4638 | . . . 4
⊢ (𝜑 → {(𝐴 +s 𝐵)} = {(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))}) | 
| 293 | 291, 292 | breqtrd 5169 | . . 3
⊢ (𝜑 → ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) <<s {(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))}) | 
| 294 |  | eqid 2737 | . . . . . . . 8
⊢ (𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) = (𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) | 
| 295 | 294 | rnmpt 5968 | . . . . . . 7
⊢ ran
(𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) = {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} | 
| 296 |  | ssltex2 27832 | . . . . . . . . . 10
⊢ (𝐿 <<s 𝑅 → 𝑅 ∈ V) | 
| 297 | 2, 296 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈ V) | 
| 298 | 297 | mptexd 7244 | . . . . . . . 8
⊢ (𝜑 → (𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) ∈ V) | 
| 299 |  | rnexg 7924 | . . . . . . . 8
⊢ ((𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) ∈ V → ran (𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) ∈ V) | 
| 300 | 298, 299 | syl 17 | . . . . . . 7
⊢ (𝜑 → ran (𝑟 ∈ 𝑅 ↦ (𝑟 +s 𝐵)) ∈ V) | 
| 301 | 295, 300 | eqeltrrid 2846 | . . . . . 6
⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∈ V) | 
| 302 |  | eqid 2737 | . . . . . . . 8
⊢ (𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) = (𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) | 
| 303 | 302 | rnmpt 5968 | . . . . . . 7
⊢ ran
(𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) = {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} | 
| 304 |  | ssltex2 27832 | . . . . . . . . . 10
⊢ (𝑀 <<s 𝑆 → 𝑆 ∈ V) | 
| 305 | 6, 304 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ V) | 
| 306 | 305 | mptexd 7244 | . . . . . . . 8
⊢ (𝜑 → (𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) ∈ V) | 
| 307 |  | rnexg 7924 | . . . . . . . 8
⊢ ((𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) ∈ V → ran (𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) ∈ V) | 
| 308 | 306, 307 | syl 17 | . . . . . . 7
⊢ (𝜑 → ran (𝑠 ∈ 𝑆 ↦ (𝐴 +s 𝑠)) ∈ V) | 
| 309 | 303, 308 | eqeltrrid 2846 | . . . . . 6
⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} ∈ V) | 
| 310 | 301, 309 | unexd 7774 | . . . . 5
⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) ∈ V) | 
| 311 | 113 | sselda 3983 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈  No
) | 
| 312 | 8 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐵 ∈  No
) | 
| 313 | 311, 312 | addscld 28013 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝑟 +s 𝐵) ∈  No
) | 
| 314 |  | eleq1 2829 | . . . . . . . . 9
⊢ (𝑤 = (𝑟 +s 𝐵) → (𝑤 ∈  No 
↔ (𝑟 +s
𝐵) ∈  No )) | 
| 315 | 313, 314 | syl5ibrcom 247 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝑤 = (𝑟 +s 𝐵) → 𝑤 ∈  No
)) | 
| 316 | 315 | rexlimdva 3155 | . . . . . . 7
⊢ (𝜑 → (∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵) → 𝑤 ∈  No
)) | 
| 317 | 316 | abssdv 4068 | . . . . . 6
⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ⊆  No
) | 
| 318 | 4 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝐴 ∈  No
) | 
| 319 | 144 | sselda 3983 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈  No
) | 
| 320 | 318, 319 | addscld 28013 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝐴 +s 𝑠) ∈  No
) | 
| 321 |  | eleq1 2829 | . . . . . . . . 9
⊢ (𝑡 = (𝐴 +s 𝑠) → (𝑡 ∈  No 
↔ (𝐴 +s
𝑠) ∈  No )) | 
| 322 | 320, 321 | syl5ibrcom 247 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑡 = (𝐴 +s 𝑠) → 𝑡 ∈  No
)) | 
| 323 | 322 | rexlimdva 3155 | . . . . . . 7
⊢ (𝜑 → (∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠) → 𝑡 ∈  No
)) | 
| 324 | 323 | abssdv 4068 | . . . . . 6
⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} ⊆  No
) | 
| 325 | 317, 324 | unssd 4192 | . . . . 5
⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) ⊆  No
) | 
| 326 |  | velsn 4642 | . . . . . . 7
⊢ (𝑎 ∈ {(𝐴 +s 𝐵)} ↔ 𝑎 = (𝐴 +s 𝐵)) | 
| 327 |  | elun 4153 | . . . . . . . . . . . . 13
⊢ (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) ↔ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∨ 𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) | 
| 328 |  | vex 3484 | . . . . . . . . . . . . . . 15
⊢ 𝑏 ∈ V | 
| 329 | 328, 125 | elab 3679 | . . . . . . . . . . . . . 14
⊢ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ↔ ∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵)) | 
| 330 | 328, 156 | elab 3679 | . . . . . . . . . . . . . 14
⊢ (𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} ↔ ∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠)) | 
| 331 | 329, 330 | orbi12i 915 | . . . . . . . . . . . . 13
⊢ ((𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∨ 𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) ↔ (∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∨ ∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠))) | 
| 332 | 327, 331 | bitri 275 | . . . . . . . . . . . 12
⊢ (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) ↔ (∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∨ ∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠))) | 
| 333 | 1 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐴 = (𝐿 |s 𝑅)) | 
| 334 | 251 | simp3d 1145 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅) | 
| 335 | 334 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → {(𝐿 |s 𝑅)} <<s 𝑅) | 
| 336 | 256 | a1i 11 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)}) | 
| 337 |  | simpr 484 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ 𝑅) | 
| 338 | 335, 336,
337 | ssltsepcd 27839 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐿 |s 𝑅) <s 𝑟) | 
| 339 | 333, 338 | eqbrtrd 5165 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐴 <s 𝑟) | 
| 340 | 4 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐴 ∈  No
) | 
| 341 | 340, 311,
312 | sltadd1d 28031 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐴 <s 𝑟 ↔ (𝐴 +s 𝐵) <s (𝑟 +s 𝐵))) | 
| 342 | 339, 341 | mpbid 232 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐴 +s 𝐵) <s (𝑟 +s 𝐵)) | 
| 343 |  | breq2 5147 | . . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝑟 +s 𝐵) → ((𝐴 +s 𝐵) <s 𝑏 ↔ (𝐴 +s 𝐵) <s (𝑟 +s 𝐵))) | 
| 344 | 342, 343 | syl5ibrcom 247 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝑏 = (𝑟 +s 𝐵) → (𝐴 +s 𝐵) <s 𝑏)) | 
| 345 | 344 | rexlimdva 3155 | . . . . . . . . . . . . 13
⊢ (𝜑 → (∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) → (𝐴 +s 𝐵) <s 𝑏)) | 
| 346 | 5 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝐵 = (𝑀 |s 𝑆)) | 
| 347 | 268 | simp3d 1145 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → {(𝑀 |s 𝑆)} <<s 𝑆) | 
| 348 | 347 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → {(𝑀 |s 𝑆)} <<s 𝑆) | 
| 349 | 273 | a1i 11 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)}) | 
| 350 |  | simpr 484 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆) | 
| 351 | 348, 349,
350 | ssltsepcd 27839 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑀 |s 𝑆) <s 𝑠) | 
| 352 | 346, 351 | eqbrtrd 5165 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝐵 <s 𝑠) | 
| 353 | 8 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝐵 ∈  No
) | 
| 354 | 353, 319,
318 | sltadd2d 28030 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝐵 <s 𝑠 ↔ (𝐴 +s 𝐵) <s (𝐴 +s 𝑠))) | 
| 355 | 352, 354 | mpbid 232 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝐴 +s 𝐵) <s (𝐴 +s 𝑠)) | 
| 356 |  | breq2 5147 | . . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝐴 +s 𝑠) → ((𝐴 +s 𝐵) <s 𝑏 ↔ (𝐴 +s 𝐵) <s (𝐴 +s 𝑠))) | 
| 357 | 355, 356 | syl5ibrcom 247 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑏 = (𝐴 +s 𝑠) → (𝐴 +s 𝐵) <s 𝑏)) | 
| 358 | 357 | rexlimdva 3155 | . . . . . . . . . . . . 13
⊢ (𝜑 → (∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠) → (𝐴 +s 𝐵) <s 𝑏)) | 
| 359 | 345, 358 | jaod 860 | . . . . . . . . . . . 12
⊢ (𝜑 → ((∃𝑟 ∈ 𝑅 𝑏 = (𝑟 +s 𝐵) ∨ ∃𝑠 ∈ 𝑆 𝑏 = (𝐴 +s 𝑠)) → (𝐴 +s 𝐵) <s 𝑏)) | 
| 360 | 332, 359 | biimtrid 242 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → (𝐴 +s 𝐵) <s 𝑏)) | 
| 361 | 360 | imp 406 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) → (𝐴 +s 𝐵) <s 𝑏) | 
| 362 |  | breq1 5146 | . . . . . . . . . 10
⊢ (𝑎 = (𝐴 +s 𝐵) → (𝑎 <s 𝑏 ↔ (𝐴 +s 𝐵) <s 𝑏)) | 
| 363 | 361, 362 | syl5ibrcom 247 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) → (𝑎 = (𝐴 +s 𝐵) → 𝑎 <s 𝑏)) | 
| 364 | 363 | ex 412 | . . . . . . . 8
⊢ (𝜑 → (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → (𝑎 = (𝐴 +s 𝐵) → 𝑎 <s 𝑏))) | 
| 365 | 364 | com23 86 | . . . . . . 7
⊢ (𝜑 → (𝑎 = (𝐴 +s 𝐵) → (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → 𝑎 <s 𝑏))) | 
| 366 | 326, 365 | biimtrid 242 | . . . . . 6
⊢ (𝜑 → (𝑎 ∈ {(𝐴 +s 𝐵)} → (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) → 𝑎 <s 𝑏))) | 
| 367 | 366 | 3imp 1111 | . . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ {(𝐴 +s 𝐵)} ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) → 𝑎 <s 𝑏) | 
| 368 | 221, 310,
238, 325, 367 | ssltd 27836 | . . . 4
⊢ (𝜑 → {(𝐴 +s 𝐵)} <<s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) | 
| 369 | 292, 368 | eqbrtrrd 5167 | . . 3
⊢ (𝜑 → {(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)}))} <<s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) | 
| 370 | 18, 110, 202, 293, 369 | cofcut1d 27955 | . 2
⊢ (𝜑 → (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)𝑎 = (𝑝 +s 𝐵)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( L ‘𝐵)𝑏 = (𝐴 +s 𝑞)}) |s ({𝑐 ∣ ∃𝑒 ∈ ( R ‘𝐴)𝑐 = (𝑒 +s 𝐵)} ∪ {𝑑 ∣ ∃𝑓 ∈ ( R ‘𝐵)𝑑 = (𝐴 +s 𝑓)})) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}))) | 
| 371 | 10, 370 | eqtrd 2777 | 1
⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}))) |