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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 4231 | . . . 4 ⊢ (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴) | |
3 | 1, 2 | sylib 218 | . . 3 ⊢ (𝜑 → (ℂ ∩ 𝐴) = 𝐴) |
4 | 3 | raleqdv 3324 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥))) |
5 | 0cn 11251 | . . . . . 6 ⊢ 0 ∈ ℂ | |
6 | fnconstg 6797 | . . . . . 6 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ (ℂ × {0}) Fn ℂ |
8 | df-0p 25719 | . . . . . 6 ⊢ 0𝑝 = (ℂ × {0}) | |
9 | 8 | fneq1i 6666 | . . . . 5 ⊢ (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ) |
10 | 7, 9 | mpbir 231 | . . . 4 ⊢ 0𝑝 Fn ℂ |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → 0𝑝 Fn ℂ) |
12 | 0pledm.2 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
13 | cnex 11234 | . . . 4 ⊢ ℂ ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → ℂ ∈ V) |
15 | ssexg 5329 | . . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V) | |
16 | 1, 13, 15 | sylancl 586 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
17 | eqid 2735 | . . 3 ⊢ (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴) | |
18 | 0pval 25720 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
19 | 18 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) |
20 | eqidd 2736 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
21 | 11, 12, 14, 16, 17, 19, 20 | ofrfval 7707 | . 2 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥))) |
22 | fnconstg 6797 | . . . . 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 4235 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
26 | c0ex 11253 | . . . . 5 ⊢ 0 ∈ V | |
27 | 26 | fvconst2 7224 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0) |
28 | 27 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0) |
29 | 24, 12, 16, 16, 25, 28, 20 | ofrfval 7707 | . 2 ⊢ (𝜑 → ((𝐴 × {0}) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥))) |
30 | 4, 21, 29 | 3bitr4d 311 | 1 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 {csn 4631 class class class wbr 5148 × cxp 5687 Fn wfn 6558 ‘cfv 6563 ∘r cofr 7696 ℂcc 11151 0cc0 11153 ≤ cle 11294 0𝑝c0p 25718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-cnex 11209 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ofr 7698 df-0p 25719 |
This theorem is referenced by: xrge0f 25781 itg20 25787 itg2const 25790 i1fibl 25858 itgitg1 25859 ftc1anclem5 37684 ftc1anclem7 37686 |
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