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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 4176 | . . . 4 ⊢ (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴) | |
3 | 1, 2 | sylib 217 | . . 3 ⊢ (𝜑 → (ℂ ∩ 𝐴) = 𝐴) |
4 | 3 | raleqdv 3314 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥))) |
5 | 0cn 11148 | . . . . . 6 ⊢ 0 ∈ ℂ | |
6 | fnconstg 6731 | . . . . . 6 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ (ℂ × {0}) Fn ℂ |
8 | df-0p 25037 | . . . . . 6 ⊢ 0𝑝 = (ℂ × {0}) | |
9 | 8 | fneq1i 6600 | . . . . 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 11133 | . . . 4 ⊢ ℂ ∈ V | |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → ℂ ∈ V) |
15 | ssexg 5281 | . . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V) | |
16 | 1, 13, 15 | sylancl 587 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
17 | eqid 2737 | . . 3 ⊢ (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴) | |
18 | 0pval 25038 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0) | |
19 | 18 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0) |
20 | eqidd 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
21 | 11, 12, 14, 16, 17, 19, 20 | ofrfval 7628 | . 2 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥))) |
22 | fnconstg 6731 | . . . . 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 4179 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
26 | c0ex 11150 | . . . . 5 ⊢ 0 ∈ V | |
27 | 26 | fvconst2 7154 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0) |
28 | 27 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0) |
29 | 24, 12, 16, 16, 25, 28, 20 | ofrfval 7628 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 Vcvv 3446 ∩ cin 3910 ⊆ wss 3911 {csn 4587 class class class wbr 5106 × cxp 5632 Fn wfn 6492 ‘cfv 6497 ∘r cofr 7617 ℂcc 11050 0cc0 11052 ≤ cle 11191 0𝑝c0p 25036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-cnex 11108 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-mulcl 11114 ax-i2m1 11120 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ofr 7619 df-0p 25037 |
This theorem is referenced by: xrge0f 25099 itg20 25105 itg2const 25108 i1fibl 25175 itgitg1 25176 ftc1anclem5 36158 ftc1anclem7 36160 |
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