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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 4216 | . . . 4 ⊢ (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴) | |
3 | 1, 2 | sylib 217 | . . 3 ⊢ (𝜑 → (ℂ ∩ 𝐴) = 𝐴) |
4 | 3 | raleqdv 3315 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥))) |
5 | 0cn 11256 | . . . . . 6 ⊢ 0 ∈ ℂ | |
6 | fnconstg 6790 | . . . . . 6 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ (ℂ × {0}) Fn ℂ |
8 | df-0p 25690 | . . . . . 6 ⊢ 0𝑝 = (ℂ × {0}) | |
9 | 8 | fneq1i 6657 | . . . . 5 ⊢ (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ) |
10 | 7, 9 | mpbir 230 | . . . 4 ⊢ 0𝑝 Fn ℂ |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → 0𝑝 Fn ℂ) |
12 | 0pledm.2 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
13 | cnex 11239 | . . . 4 ⊢ ℂ ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → ℂ ∈ V) |
15 | ssexg 5328 | . . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V) | |
16 | 1, 13, 15 | sylancl 584 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
17 | eqid 2726 | . . 3 ⊢ (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴) | |
18 | 0pval 25691 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
19 | 18 | adantl 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) |
20 | eqidd 2727 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
21 | 11, 12, 14, 16, 17, 19, 20 | ofrfval 7700 | . 2 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥))) |
22 | fnconstg 6790 | . . . . 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 4220 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
26 | c0ex 11258 | . . . . 5 ⊢ 0 ∈ V | |
27 | 26 | fvconst2 7221 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0) |
28 | 27 | adantl 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0) |
29 | 24, 12, 16, 16, 25, 28, 20 | ofrfval 7700 | . 2 ⊢ (𝜑 → ((𝐴 × {0}) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥))) |
30 | 4, 21, 29 | 3bitr4d 310 | 1 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ∩ cin 3946 ⊆ wss 3947 {csn 4633 class class class wbr 5153 × cxp 5680 Fn wfn 6549 ‘cfv 6554 ∘r cofr 7689 ℂcc 11156 0cc0 11158 ≤ cle 11299 0𝑝c0p 25689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-cnex 11214 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-mulcl 11220 ax-i2m1 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ofr 7691 df-0p 25690 |
This theorem is referenced by: xrge0f 25752 itg20 25758 itg2const 25761 i1fibl 25828 itgitg1 25829 ftc1anclem5 37398 ftc1anclem7 37400 |
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