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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 4222 | . . . 4 ⊢ (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴) | |
| 3 | 1, 2 | sylib 218 | . . 3 ⊢ (𝜑 → (ℂ ∩ 𝐴) = 𝐴) | 
| 4 | 3 | raleqdv 3325 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥))) | 
| 5 | 0cn 11254 | . . . . . 6 ⊢ 0 ∈ ℂ | |
| 6 | fnconstg 6795 | . . . . . 6 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ) | |
| 7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ (ℂ × {0}) Fn ℂ | 
| 8 | df-0p 25706 | . . . . . 6 ⊢ 0𝑝 = (ℂ × {0}) | |
| 9 | 8 | fneq1i 6664 | . . . . 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 11237 | . . . 4 ⊢ ℂ ∈ V | |
| 14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → ℂ ∈ V) | 
| 15 | ssexg 5322 | . . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V) | |
| 16 | 1, 13, 15 | sylancl 586 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | 
| 17 | eqid 2736 | . . 3 ⊢ (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴) | |
| 18 | 0pval 25707 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
| 19 | 18 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) | 
| 20 | eqidd 2737 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
| 21 | 11, 12, 14, 16, 17, 19, 20 | ofrfval 7708 | . 2 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥))) | 
| 22 | fnconstg 6795 | . . . . 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 4226 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
| 26 | c0ex 11256 | . . . . 5 ⊢ 0 ∈ V | |
| 27 | 26 | fvconst2 7225 | . . . 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 7708 | . 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 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 {csn 4625 class class class wbr 5142 × cxp 5682 Fn wfn 6555 ‘cfv 6560 ∘r cofr 7697 ℂcc 11154 0cc0 11156 ≤ cle 11297 0𝑝c0p 25705 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-cnex 11212 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-mulcl 11218 ax-i2m1 11224 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ofr 7699 df-0p 25706 | 
| This theorem is referenced by: xrge0f 25767 itg20 25773 itg2const 25776 i1fibl 25844 itgitg1 25845 ftc1anclem5 37705 ftc1anclem7 37707 | 
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