| Step | Hyp | Ref
| Expression |
| 1 | | suplocsrlem.ss |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ⊆ R) |
| 2 | | suplocsrlem.c |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ 𝐴) |
| 3 | 1, 2 | sseldd 3184 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ R) |
| 4 | | 0idsr 7834 |
. . . . . . 7
⊢ (𝐶 ∈ R →
(𝐶
+R 0R) = 𝐶) |
| 5 | 3, 4 | syl 14 |
. . . . . 6
⊢ (𝜑 → (𝐶 +R
0R) = 𝐶) |
| 6 | 5, 2 | eqeltrd 2273 |
. . . . 5
⊢ (𝜑 → (𝐶 +R
0R) ∈ 𝐴) |
| 7 | | 1pr 7621 |
. . . . 5
⊢
1P ∈ P |
| 8 | 6, 7 | jctil 312 |
. . . 4
⊢ (𝜑 →
(1P ∈ P ∧ (𝐶 +R
0R) ∈ 𝐴)) |
| 9 | | opeq1 3808 |
. . . . . . . . 9
⊢ (𝑤 = 1P
→ 〈𝑤,
1P〉 = 〈1P,
1P〉) |
| 10 | 9 | eceq1d 6628 |
. . . . . . . 8
⊢ (𝑤 = 1P
→ [〈𝑤,
1P〉] ~R =
[〈1P, 1P〉]
~R ) |
| 11 | | df-0r 7798 |
. . . . . . . 8
⊢
0R = [〈1P,
1P〉] ~R |
| 12 | 10, 11 | eqtr4di 2247 |
. . . . . . 7
⊢ (𝑤 = 1P
→ [〈𝑤,
1P〉] ~R =
0R) |
| 13 | 12 | oveq2d 5938 |
. . . . . 6
⊢ (𝑤 = 1P
→ (𝐶
+R [〈𝑤, 1P〉]
~R ) = (𝐶 +R
0R)) |
| 14 | 13 | eleq1d 2265 |
. . . . 5
⊢ (𝑤 = 1P
→ ((𝐶
+R [〈𝑤, 1P〉]
~R ) ∈ 𝐴 ↔ (𝐶 +R
0R) ∈ 𝐴)) |
| 15 | | suplocsrlem.b |
. . . . 5
⊢ 𝐵 = {𝑤 ∈ P ∣ (𝐶 +R
[〈𝑤,
1P〉] ~R ) ∈ 𝐴} |
| 16 | 14, 15 | elrab2 2923 |
. . . 4
⊢
(1P ∈ 𝐵 ↔ (1P ∈
P ∧ (𝐶
+R 0R) ∈ 𝐴)) |
| 17 | 8, 16 | sylibr 134 |
. . 3
⊢ (𝜑 →
1P ∈ 𝐵) |
| 18 | | elex2 2779 |
. . 3
⊢
(1P ∈ 𝐵 → ∃𝑣 𝑣 ∈ 𝐵) |
| 19 | 17, 18 | syl 14 |
. 2
⊢ (𝜑 → ∃𝑣 𝑣 ∈ 𝐵) |
| 20 | | suplocsrlem.ub |
. . . 4
⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) |
| 21 | | breq1 4036 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐶 → (𝑦 <R 𝑥 ↔ 𝐶 <R 𝑥)) |
| 22 | 21 | rspccv 2865 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
𝐴 𝑦 <R 𝑥 → (𝐶 ∈ 𝐴 → 𝐶 <R 𝑥)) |
| 23 | 2, 22 | mpan9 281 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → 𝐶 <R 𝑥) |
| 24 | | 0lt1sr 7832 |
. . . . . . . . . . . . . 14
⊢
0R <R
1R |
| 25 | | 0r 7817 |
. . . . . . . . . . . . . . 15
⊢
0R ∈ R |
| 26 | | 1sr 7818 |
. . . . . . . . . . . . . . 15
⊢
1R ∈ R |
| 27 | | m1r 7819 |
. . . . . . . . . . . . . . 15
⊢
-1R ∈ R |
| 28 | | ltasrg 7837 |
. . . . . . . . . . . . . . 15
⊢
((0R ∈ R ∧
1R ∈ R ∧
-1R ∈ R) →
(0R <R
1R ↔ (-1R
+R 0R)
<R (-1R
+R 1R))) |
| 29 | 25, 26, 27, 28 | mp3an 1348 |
. . . . . . . . . . . . . 14
⊢
(0R <R
1R ↔ (-1R
+R 0R)
<R (-1R
+R 1R)) |
| 30 | 24, 29 | mpbi 145 |
. . . . . . . . . . . . 13
⊢
(-1R +R
0R) <R
(-1R +R
1R) |
| 31 | | 0idsr 7834 |
. . . . . . . . . . . . . 14
⊢
(-1R ∈ R →
(-1R +R
0R) = -1R) |
| 32 | 27, 31 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(-1R +R
0R) = -1R |
| 33 | | m1p1sr 7827 |
. . . . . . . . . . . . 13
⊢
(-1R +R
1R) = 0R |
| 34 | 30, 32, 33 | 3brtr3i 4062 |
. . . . . . . . . . . 12
⊢
-1R <R
0R |
| 35 | | ltasrg 7837 |
. . . . . . . . . . . . 13
⊢
((-1R ∈ R ∧
0R ∈ R ∧ 𝐶 ∈ R) →
(-1R <R
0R ↔ (𝐶 +R
-1R) <R (𝐶 +R
0R))) |
| 36 | 27, 25, 3, 35 | mp3an12i 1352 |
. . . . . . . . . . . 12
⊢ (𝜑 →
(-1R <R
0R ↔ (𝐶 +R
-1R) <R (𝐶 +R
0R))) |
| 37 | 34, 36 | mpbii 148 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐶 +R
-1R) <R (𝐶 +R
0R)) |
| 38 | 37, 5 | breqtrd 4059 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐶 +R
-1R) <R 𝐶) |
| 39 | | ltsosr 7831 |
. . . . . . . . . . 11
⊢
<R Or R |
| 40 | | ltrelsr 7805 |
. . . . . . . . . . 11
⊢
<R ⊆ (R ×
R) |
| 41 | 39, 40 | sotri 5065 |
. . . . . . . . . 10
⊢ (((𝐶 +R
-1R) <R 𝐶 ∧ 𝐶 <R 𝑥) → (𝐶 +R
-1R) <R 𝑥) |
| 42 | 38, 41 | sylan 283 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐶 <R 𝑥) → (𝐶 +R
-1R) <R 𝑥) |
| 43 | | map2psrprg 7872 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ R →
((𝐶
+R -1R)
<R 𝑥 ↔ ∃𝑣 ∈ P (𝐶 +R [〈𝑣,
1P〉] ~R ) = 𝑥)) |
| 44 | 3, 43 | syl 14 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐶 +R
-1R) <R 𝑥 ↔ ∃𝑣 ∈ P (𝐶 +R [〈𝑣,
1P〉] ~R ) = 𝑥)) |
| 45 | 44 | adantr 276 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐶 <R 𝑥) → ((𝐶 +R
-1R) <R 𝑥 ↔ ∃𝑣 ∈ P (𝐶 +R [〈𝑣,
1P〉] ~R ) = 𝑥)) |
| 46 | 42, 45 | mpbid 147 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐶 <R 𝑥) → ∃𝑣 ∈ P (𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥) |
| 47 | 23, 46 | syldan 282 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑣 ∈ P (𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥) |
| 48 | | breq1 4036 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐶 +R [〈𝑤,
1P〉] ~R ) → (𝑦 <R
(𝐶
+R [〈𝑣, 1P〉]
~R ) ↔ (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R
))) |
| 49 | | simpllr 534 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) ∧ (𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥) → ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) |
| 50 | | breq2 4037 |
. . . . . . . . . . . . . . 15
⊢ ((𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥 → (𝑦 <R (𝐶 +R
[〈𝑣,
1P〉] ~R ) ↔ 𝑦 <R
𝑥)) |
| 51 | 50 | ralbidv 2497 |
. . . . . . . . . . . . . 14
⊢ ((𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R
[〈𝑣,
1P〉] ~R ) ↔
∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)) |
| 52 | 51 | adantl 277 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) ∧ (𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥) → (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R
[〈𝑣,
1P〉] ~R ) ↔
∀𝑦 ∈ 𝐴 𝑦 <R 𝑥)) |
| 53 | 49, 52 | mpbird 167 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) ∧ (𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥) → ∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R
[〈𝑣,
1P〉] ~R
)) |
| 54 | 53 | adantr 276 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧
∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) ∧ (𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥) ∧ 𝑤 ∈ 𝐵) → ∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R
[〈𝑣,
1P〉] ~R
)) |
| 55 | 15 | rabeq2i 2760 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈ 𝐵 ↔ (𝑤 ∈ P ∧ (𝐶 +R
[〈𝑤,
1P〉] ~R ) ∈ 𝐴)) |
| 56 | 55 | simprbi 275 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ 𝐵 → (𝐶 +R [〈𝑤,
1P〉] ~R ) ∈ 𝐴) |
| 57 | 56 | adantl 277 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧
∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) ∧ (𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥) ∧ 𝑤 ∈ 𝐵) → (𝐶 +R [〈𝑤,
1P〉] ~R ) ∈ 𝐴) |
| 58 | 48, 54, 57 | rspcdva 2873 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧
∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) ∧ (𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥) ∧ 𝑤 ∈ 𝐵) → (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R
)) |
| 59 | 58 | ralrimiva 2570 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) ∧ (𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥) → ∀𝑤 ∈ 𝐵 (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R
)) |
| 60 | 59 | ex 115 |
. . . . . . . 8
⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) → ((𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥 → ∀𝑤 ∈ 𝐵 (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R
))) |
| 61 | 60 | reximdva 2599 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → (∃𝑣 ∈ P (𝐶 +R
[〈𝑣,
1P〉] ~R ) = 𝑥 → ∃𝑣 ∈ P
∀𝑤 ∈ 𝐵 (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R
))) |
| 62 | 47, 61 | mpd 13 |
. . . . . 6
⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑣 ∈ P
∀𝑤 ∈ 𝐵 (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R
)) |
| 63 | 62 | ex 115 |
. . . . 5
⊢ (𝜑 → (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → ∃𝑣 ∈ P
∀𝑤 ∈ 𝐵 (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R
))) |
| 64 | 63 | rexlimdvw 2618 |
. . . 4
⊢ (𝜑 → (∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → ∃𝑣 ∈ P
∀𝑤 ∈ 𝐵 (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R
))) |
| 65 | 20, 64 | mpd 13 |
. . 3
⊢ (𝜑 → ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R
)) |
| 66 | | elrabi 2917 |
. . . . . . . 8
⊢ (𝑤 ∈ {𝑎 ∈ P ∣ (𝐶 +R
[〈𝑎,
1P〉] ~R ) ∈ 𝐴} → 𝑤 ∈ P) |
| 67 | | opeq1 3808 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑎 → 〈𝑤, 1P〉 =
〈𝑎,
1P〉) |
| 68 | 67 | eceq1d 6628 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑎 → [〈𝑤, 1P〉]
~R = [〈𝑎, 1P〉]
~R ) |
| 69 | 68 | oveq2d 5938 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑎 → (𝐶 +R [〈𝑤,
1P〉] ~R ) = (𝐶 +R
[〈𝑎,
1P〉] ~R
)) |
| 70 | 69 | eleq1d 2265 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑎 → ((𝐶 +R [〈𝑤,
1P〉] ~R ) ∈ 𝐴 ↔ (𝐶 +R [〈𝑎,
1P〉] ~R ) ∈ 𝐴)) |
| 71 | 70 | cbvrabv 2762 |
. . . . . . . . 9
⊢ {𝑤 ∈ P ∣
(𝐶
+R [〈𝑤, 1P〉]
~R ) ∈ 𝐴} = {𝑎 ∈ P ∣ (𝐶 +R
[〈𝑎,
1P〉] ~R ) ∈ 𝐴} |
| 72 | 15, 71 | eqtri 2217 |
. . . . . . . 8
⊢ 𝐵 = {𝑎 ∈ P ∣ (𝐶 +R
[〈𝑎,
1P〉] ~R ) ∈ 𝐴} |
| 73 | 66, 72 | eleq2s 2291 |
. . . . . . 7
⊢ (𝑤 ∈ 𝐵 → 𝑤 ∈ P) |
| 74 | 73 | adantl 277 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑣 ∈ P) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ P) |
| 75 | | simplr 528 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑣 ∈ P) ∧ 𝑤 ∈ 𝐵) → 𝑣 ∈ P) |
| 76 | 3 | ad2antrr 488 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑣 ∈ P) ∧ 𝑤 ∈ 𝐵) → 𝐶 ∈ R) |
| 77 | | ltpsrprg 7870 |
. . . . . 6
⊢ ((𝑤 ∈ P ∧
𝑣 ∈ P
∧ 𝐶 ∈
R) → ((𝐶
+R [〈𝑤, 1P〉]
~R ) <R (𝐶 +R [〈𝑣,
1P〉] ~R ) ↔ 𝑤<P
𝑣)) |
| 78 | 74, 75, 76, 77 | syl3anc 1249 |
. . . . 5
⊢ (((𝜑 ∧ 𝑣 ∈ P) ∧ 𝑤 ∈ 𝐵) → ((𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R ) ↔ 𝑤<P
𝑣)) |
| 79 | 78 | ralbidva 2493 |
. . . 4
⊢ ((𝜑 ∧ 𝑣 ∈ P) →
(∀𝑤 ∈ 𝐵 (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R ) ↔
∀𝑤 ∈ 𝐵 𝑤<P 𝑣)) |
| 80 | 79 | rexbidva 2494 |
. . 3
⊢ (𝜑 → (∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 (𝐶 +R [〈𝑤,
1P〉] ~R )
<R (𝐶 +R [〈𝑣,
1P〉] ~R ) ↔
∃𝑣 ∈
P ∀𝑤
∈ 𝐵 𝑤<P 𝑣)) |
| 81 | 65, 80 | mpbid 147 |
. 2
⊢ (𝜑 → ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 𝑤<P 𝑣) |
| 82 | | suplocsrlem.loc |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R
𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) |
| 83 | 15, 1, 2, 20, 82 | suplocsrlemb 7873 |
. 2
⊢ (𝜑 → ∀𝑣 ∈ P ∀𝑤 ∈ P (𝑣<P
𝑤 → (∃𝑢 ∈ 𝐵 𝑣<P 𝑢 ∨ ∀𝑢 ∈ 𝐵 𝑢<P 𝑤))) |
| 84 | 19, 81, 83 | suplocexpr 7792 |
1
⊢ (𝜑 → ∃𝑣 ∈ P (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤 ∈ P (𝑤<P
𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢))) |