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Mirrors > Home > MPE Home > Th. List > 0pledm | Structured version Visualization version GIF version |
Description: Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
0pledm.1 | ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
0pledm.2 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
Ref | Expression |
---|---|
0pledm | ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0pledm.1 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
2 | sseqin2 4189 | . . . 4 ⊢ (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴) | |
3 | 1, 2 | sylib 219 | . . 3 ⊢ (𝜑 → (ℂ ∩ 𝐴) = 𝐴) |
4 | 3 | raleqdv 3413 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥))) |
5 | 0cn 10621 | . . . . . 6 ⊢ 0 ∈ ℂ | |
6 | fnconstg 6560 | . . . . . 6 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ (ℂ × {0}) Fn ℂ |
8 | df-0p 24198 | . . . . . 6 ⊢ 0𝑝 = (ℂ × {0}) | |
9 | 8 | fneq1i 6443 | . . . . 5 ⊢ (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ) |
10 | 7, 9 | mpbir 232 | . . . 4 ⊢ 0𝑝 Fn ℂ |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → 0𝑝 Fn ℂ) |
12 | 0pledm.2 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
13 | cnex 10606 | . . . 4 ⊢ ℂ ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → ℂ ∈ V) |
15 | ssexg 5218 | . . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V) | |
16 | 1, 13, 15 | sylancl 586 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
17 | eqid 2818 | . . 3 ⊢ (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴) | |
18 | 0pval 24199 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
19 | 18 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) |
20 | eqidd 2819 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
21 | 11, 12, 14, 16, 17, 19, 20 | ofrfval 7406 | . 2 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥))) |
22 | fnconstg 6560 | . . . . 5 ⊢ (0 ∈ ℂ → (𝐴 × {0}) Fn 𝐴) | |
23 | 5, 22 | ax-mp 5 | . . . 4 ⊢ (𝐴 × {0}) Fn 𝐴 |
24 | 23 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {0}) Fn 𝐴) |
25 | inidm 4192 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
26 | c0ex 10623 | . . . . 5 ⊢ 0 ∈ V | |
27 | 26 | fvconst2 6958 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0) |
28 | 27 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0) |
29 | 24, 12, 16, 16, 25, 28, 20 | ofrfval 7406 | . 2 ⊢ (𝜑 → ((𝐴 × {0}) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥))) |
30 | 4, 21, 29 | 3bitr4d 312 | 1 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 {csn 4557 class class class wbr 5057 × cxp 5546 Fn wfn 6343 ‘cfv 6348 ∘r cofr 7397 ℂcc 10523 0cc0 10525 ≤ cle 10664 0𝑝c0p 24197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-cnex 10581 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-mulcl 10587 ax-i2m1 10593 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ofr 7399 df-0p 24198 |
This theorem is referenced by: xrge0f 24259 itg20 24265 itg2const 24268 i1fibl 24335 itgitg1 24336 ftc1anclem5 34852 ftc1anclem7 34854 |
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