| Step | Hyp | Ref
| Expression |
| 1 | | hlimf 31256 |
. . . . 5
⊢
⇝𝑣 :dom ⇝𝑣 ⟶
ℋ |
| 2 | | ffun 6739 |
. . . . 5
⊢ (
⇝𝑣 :dom ⇝𝑣 ⟶ ℋ
→ Fun ⇝𝑣 ) |
| 3 | 1, 2 | ax-mp 5 |
. . . 4
⊢ Fun
⇝𝑣 |
| 4 | | chscl.5 |
. . . 4
⊢ (𝜑 → 𝐻 ⇝𝑣 𝑢) |
| 5 | | funbrfv 6957 |
. . . 4
⊢ (Fun
⇝𝑣 → (𝐻 ⇝𝑣 𝑢 → (
⇝𝑣 ‘𝐻) = 𝑢)) |
| 6 | 3, 4, 5 | mpsyl 68 |
. . 3
⊢ (𝜑 → (
⇝𝑣 ‘𝐻) = 𝑢) |
| 7 | | chscl.4 |
. . . . . . 7
⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵)) |
| 8 | 7 | feqmptd 6977 |
. . . . . 6
⊢ (𝜑 → 𝐻 = (𝑘 ∈ ℕ ↦ (𝐻‘𝑘))) |
| 9 | 7 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) ∈ (𝐴 +ℋ 𝐵)) |
| 10 | | chscl.1 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ Cℋ
) |
| 11 | | chsh 31243 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈
Cℋ → 𝐴 ∈ Sℋ
) |
| 12 | 10, 11 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ Sℋ
) |
| 13 | | chscl.2 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ∈ Cℋ
) |
| 14 | | chsh 31243 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈
Cℋ → 𝐵 ∈ Sℋ
) |
| 15 | 13, 14 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ∈ Sℋ
) |
| 16 | | shsel 31333 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈
Sℋ ∧ 𝐵 ∈ Sℋ )
→ ((𝐻‘𝑘) ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐻‘𝑘) = (𝑥 +ℎ 𝑦))) |
| 17 | 12, 15, 16 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐻‘𝑘) ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐻‘𝑘) = (𝑥 +ℎ 𝑦))) |
| 18 | 17 | biimpa 476 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐻‘𝑘) ∈ (𝐴 +ℋ 𝐵)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) |
| 19 | 9, 18 | syldan 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) |
| 20 | | simp3 1139 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) |
| 21 | | simp1l 1198 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝜑) |
| 22 | 21, 10 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝐴 ∈ Cℋ
) |
| 23 | 21, 13 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝐵 ∈ Cℋ
) |
| 24 | | chscl.3 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) |
| 25 | 21, 24 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝐵 ⊆ (⊥‘𝐴)) |
| 26 | 21, 7 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵)) |
| 27 | 21, 4 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝐻 ⇝𝑣 𝑢) |
| 28 | | chscl.6 |
. . . . . . . . . . . . 13
⊢ 𝐹 = (𝑛 ∈ ℕ ↦
((projℎ‘𝐴)‘(𝐻‘𝑛))) |
| 29 | | simp1r 1199 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝑘 ∈ ℕ) |
| 30 | | simp2l 1200 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝑥 ∈ 𝐴) |
| 31 | | simp2r 1201 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝑦 ∈ 𝐵) |
| 32 | 22, 23, 25, 26, 27, 28, 29, 30, 31, 20 | chscllem3 31658 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝑥 = (𝐹‘𝑘)) |
| 33 | | chsscon2 31521 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐵 ∈
Cℋ ∧ 𝐴 ∈ Cℋ )
→ (𝐵 ⊆
(⊥‘𝐴) ↔
𝐴 ⊆
(⊥‘𝐵))) |
| 34 | 13, 10, 33 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐵 ⊆ (⊥‘𝐴) ↔ 𝐴 ⊆ (⊥‘𝐵))) |
| 35 | 24, 34 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 ⊆ (⊥‘𝐵)) |
| 36 | 21, 35 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝐴 ⊆ (⊥‘𝐵)) |
| 37 | | shscom 31338 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈
Sℋ ∧ 𝐵 ∈ Sℋ )
→ (𝐴
+ℋ 𝐵) =
(𝐵 +ℋ
𝐴)) |
| 38 | 12, 15, 37 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) |
| 39 | 38 | feq3d 6723 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐻:ℕ⟶(𝐴 +ℋ 𝐵) ↔ 𝐻:ℕ⟶(𝐵 +ℋ 𝐴))) |
| 40 | 7, 39 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐻:ℕ⟶(𝐵 +ℋ 𝐴)) |
| 41 | 21, 40 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝐻:ℕ⟶(𝐵 +ℋ 𝐴)) |
| 42 | | chscl.7 |
. . . . . . . . . . . . 13
⊢ 𝐺 = (𝑛 ∈ ℕ ↦
((projℎ‘𝐵)‘(𝐻‘𝑛))) |
| 43 | | shss 31229 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈
Sℋ → 𝐴 ⊆ ℋ) |
| 44 | 12, 43 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ⊆ ℋ) |
| 45 | 21, 44 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝐴 ⊆ ℋ) |
| 46 | 45, 30 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝑥 ∈ ℋ) |
| 47 | | shss 31229 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈
Sℋ → 𝐵 ⊆ ℋ) |
| 48 | 15, 47 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐵 ⊆ ℋ) |
| 49 | 21, 48 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝐵 ⊆ ℋ) |
| 50 | 49, 31 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝑦 ∈ ℋ) |
| 51 | | ax-hvcom 31020 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) |
| 52 | 46, 50, 51 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → (𝑥 +ℎ 𝑦) = (𝑦 +ℎ 𝑥)) |
| 53 | 20, 52 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → (𝐻‘𝑘) = (𝑦 +ℎ 𝑥)) |
| 54 | 23, 22, 36, 41, 27, 42, 29, 31, 30, 53 | chscllem3 31658 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → 𝑦 = (𝐺‘𝑘)) |
| 55 | 32, 54 | oveq12d 7449 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → (𝑥 +ℎ 𝑦) = ((𝐹‘𝑘) +ℎ (𝐺‘𝑘))) |
| 56 | 20, 55 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐻‘𝑘) = (𝑥 +ℎ 𝑦)) → (𝐻‘𝑘) = ((𝐹‘𝑘) +ℎ (𝐺‘𝑘))) |
| 57 | 56 | 3exp 1120 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝐻‘𝑘) = (𝑥 +ℎ 𝑦) → (𝐻‘𝑘) = ((𝐹‘𝑘) +ℎ (𝐺‘𝑘))))) |
| 58 | 57 | rexlimdvv 3212 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐻‘𝑘) = (𝑥 +ℎ 𝑦) → (𝐻‘𝑘) = ((𝐹‘𝑘) +ℎ (𝐺‘𝑘)))) |
| 59 | 19, 58 | mpd 15 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = ((𝐹‘𝑘) +ℎ (𝐺‘𝑘))) |
| 60 | 59 | mpteq2dva 5242 |
. . . . . 6
⊢ (𝜑 → (𝑘 ∈ ℕ ↦ (𝐻‘𝑘)) = (𝑘 ∈ ℕ ↦ ((𝐹‘𝑘) +ℎ (𝐺‘𝑘)))) |
| 61 | 8, 60 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘𝑘) +ℎ (𝐺‘𝑘)))) |
| 62 | 10, 13, 24, 7, 4, 28 | chscllem1 31656 |
. . . . . . 7
⊢ (𝜑 → 𝐹:ℕ⟶𝐴) |
| 63 | 62, 44 | fssd 6753 |
. . . . . 6
⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) |
| 64 | 13, 10, 35, 40, 4, 42 | chscllem1 31656 |
. . . . . . 7
⊢ (𝜑 → 𝐺:ℕ⟶𝐵) |
| 65 | 64, 48 | fssd 6753 |
. . . . . 6
⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) |
| 66 | 10, 13, 24, 7, 4, 28 | chscllem2 31657 |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ dom ⇝𝑣
) |
| 67 | | funfvbrb 7071 |
. . . . . . . 8
⊢ (Fun
⇝𝑣 → (𝐹 ∈ dom ⇝𝑣
↔ 𝐹
⇝𝑣 ( ⇝𝑣 ‘𝐹))) |
| 68 | 3, 67 | ax-mp 5 |
. . . . . . 7
⊢ (𝐹 ∈ dom
⇝𝑣 ↔ 𝐹 ⇝𝑣 (
⇝𝑣 ‘𝐹)) |
| 69 | 66, 68 | sylib 218 |
. . . . . 6
⊢ (𝜑 → 𝐹 ⇝𝑣 (
⇝𝑣 ‘𝐹)) |
| 70 | 13, 10, 35, 40, 4, 42 | chscllem2 31657 |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ dom ⇝𝑣
) |
| 71 | | funfvbrb 7071 |
. . . . . . . 8
⊢ (Fun
⇝𝑣 → (𝐺 ∈ dom ⇝𝑣
↔ 𝐺
⇝𝑣 ( ⇝𝑣 ‘𝐺))) |
| 72 | 3, 71 | ax-mp 5 |
. . . . . . 7
⊢ (𝐺 ∈ dom
⇝𝑣 ↔ 𝐺 ⇝𝑣 (
⇝𝑣 ‘𝐺)) |
| 73 | 70, 72 | sylib 218 |
. . . . . 6
⊢ (𝜑 → 𝐺 ⇝𝑣 (
⇝𝑣 ‘𝐺)) |
| 74 | | eqid 2737 |
. . . . . 6
⊢ (𝑘 ∈ ℕ ↦ ((𝐹‘𝑘) +ℎ (𝐺‘𝑘))) = (𝑘 ∈ ℕ ↦ ((𝐹‘𝑘) +ℎ (𝐺‘𝑘))) |
| 75 | 63, 65, 69, 73, 74 | hlimadd 31212 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((𝐹‘𝑘) +ℎ (𝐺‘𝑘))) ⇝𝑣 ((
⇝𝑣 ‘𝐹) +ℎ (
⇝𝑣 ‘𝐺))) |
| 76 | 61, 75 | eqbrtrd 5165 |
. . . 4
⊢ (𝜑 → 𝐻 ⇝𝑣 ((
⇝𝑣 ‘𝐹) +ℎ (
⇝𝑣 ‘𝐺))) |
| 77 | | funbrfv 6957 |
. . . 4
⊢ (Fun
⇝𝑣 → (𝐻 ⇝𝑣 ((
⇝𝑣 ‘𝐹) +ℎ (
⇝𝑣 ‘𝐺)) → ( ⇝𝑣
‘𝐻) = ((
⇝𝑣 ‘𝐹) +ℎ (
⇝𝑣 ‘𝐺)))) |
| 78 | 3, 76, 77 | mpsyl 68 |
. . 3
⊢ (𝜑 → (
⇝𝑣 ‘𝐻) = (( ⇝𝑣
‘𝐹)
+ℎ ( ⇝𝑣 ‘𝐺))) |
| 79 | 6, 78 | eqtr3d 2779 |
. 2
⊢ (𝜑 → 𝑢 = (( ⇝𝑣
‘𝐹)
+ℎ ( ⇝𝑣 ‘𝐺))) |
| 80 | | fvex 6919 |
. . . . 5
⊢ (
⇝𝑣 ‘𝐹) ∈ V |
| 81 | 80 | chlimi 31253 |
. . . 4
⊢ ((𝐴 ∈
Cℋ ∧ 𝐹:ℕ⟶𝐴 ∧ 𝐹 ⇝𝑣 (
⇝𝑣 ‘𝐹)) → ( ⇝𝑣
‘𝐹) ∈ 𝐴) |
| 82 | 10, 62, 69, 81 | syl3anc 1373 |
. . 3
⊢ (𝜑 → (
⇝𝑣 ‘𝐹) ∈ 𝐴) |
| 83 | | fvex 6919 |
. . . . 5
⊢ (
⇝𝑣 ‘𝐺) ∈ V |
| 84 | 83 | chlimi 31253 |
. . . 4
⊢ ((𝐵 ∈
Cℋ ∧ 𝐺:ℕ⟶𝐵 ∧ 𝐺 ⇝𝑣 (
⇝𝑣 ‘𝐺)) → ( ⇝𝑣
‘𝐺) ∈ 𝐵) |
| 85 | 13, 64, 73, 84 | syl3anc 1373 |
. . 3
⊢ (𝜑 → (
⇝𝑣 ‘𝐺) ∈ 𝐵) |
| 86 | | shsva 31339 |
. . . 4
⊢ ((𝐴 ∈
Sℋ ∧ 𝐵 ∈ Sℋ )
→ ((( ⇝𝑣 ‘𝐹) ∈ 𝐴 ∧ ( ⇝𝑣
‘𝐺) ∈ 𝐵) → ((
⇝𝑣 ‘𝐹) +ℎ (
⇝𝑣 ‘𝐺)) ∈ (𝐴 +ℋ 𝐵))) |
| 87 | 12, 15, 86 | syl2anc 584 |
. . 3
⊢ (𝜑 → (((
⇝𝑣 ‘𝐹) ∈ 𝐴 ∧ ( ⇝𝑣
‘𝐺) ∈ 𝐵) → ((
⇝𝑣 ‘𝐹) +ℎ (
⇝𝑣 ‘𝐺)) ∈ (𝐴 +ℋ 𝐵))) |
| 88 | 82, 85, 87 | mp2and 699 |
. 2
⊢ (𝜑 → ((
⇝𝑣 ‘𝐹) +ℎ (
⇝𝑣 ‘𝐺)) ∈ (𝐴 +ℋ 𝐵)) |
| 89 | 79, 88 | eqeltrd 2841 |
1
⊢ (𝜑 → 𝑢 ∈ (𝐴 +ℋ 𝐵)) |