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Theorem 0pledm 25709
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.)
Hypotheses
Ref Expression
0pledm.1 (𝜑𝐴 ⊆ ℂ)
0pledm.2 (𝜑𝐹 Fn 𝐴)
Assertion
Ref Expression
0pledm (𝜑 → (0𝑝r𝐹 ↔ (𝐴 × {0}) ∘r𝐹))

Proof of Theorem 0pledm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 0pledm.1 . . . 4 (𝜑𝐴 ⊆ ℂ)
2 sseqin2 4222 . . . 4 (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴)
31, 2sylib 218 . . 3 (𝜑 → (ℂ ∩ 𝐴) = 𝐴)
43raleqdv 3325 . 2 (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹𝑥) ↔ ∀𝑥𝐴 0 ≤ (𝐹𝑥)))
5 0cn 11254 . . . . . 6 0 ∈ ℂ
6 fnconstg 6795 . . . . . 6 (0 ∈ ℂ → (ℂ × {0}) Fn ℂ)
75, 6ax-mp 5 . . . . 5 (ℂ × {0}) Fn ℂ
8 df-0p 25706 . . . . . 6 0𝑝 = (ℂ × {0})
98fneq1i 6664 . . . . 5 (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ)
107, 9mpbir 231 . . . 4 0𝑝 Fn ℂ
1110a1i 11 . . 3 (𝜑 → 0𝑝 Fn ℂ)
12 0pledm.2 . . 3 (𝜑𝐹 Fn 𝐴)
13 cnex 11237 . . . 4 ℂ ∈ V
1413a1i 11 . . 3 (𝜑 → ℂ ∈ V)
15 ssexg 5322 . . . 4 ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V)
161, 13, 15sylancl 586 . . 3 (𝜑𝐴 ∈ V)
17 eqid 2736 . . 3 (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴)
18 0pval 25707 . . . 4 (𝑥 ∈ ℂ → (0𝑝𝑥) = 0)
1918adantl 481 . . 3 ((𝜑𝑥 ∈ ℂ) → (0𝑝𝑥) = 0)
20 eqidd 2737 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
2111, 12, 14, 16, 17, 19, 20ofrfval 7708 . 2 (𝜑 → (0𝑝r𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹𝑥)))
22 fnconstg 6795 . . . . 5 (0 ∈ ℂ → (𝐴 × {0}) Fn 𝐴)
235, 22ax-mp 5 . . . 4 (𝐴 × {0}) Fn 𝐴
2423a1i 11 . . 3 (𝜑 → (𝐴 × {0}) Fn 𝐴)
25 inidm 4226 . . 3 (𝐴𝐴) = 𝐴
26 c0ex 11256 . . . . 5 0 ∈ V
2726fvconst2 7225 . . . 4 (𝑥𝐴 → ((𝐴 × {0})‘𝑥) = 0)
2827adantl 481 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {0})‘𝑥) = 0)
2924, 12, 16, 16, 25, 28, 20ofrfval 7708 . 2 (𝜑 → ((𝐴 × {0}) ∘r𝐹 ↔ ∀𝑥𝐴 0 ≤ (𝐹𝑥)))
304, 21, 293bitr4d 311 1 (𝜑 → (0𝑝r𝐹 ↔ (𝐴 × {0}) ∘r𝐹))
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  cc 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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