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Theorem 0pledm 23380
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 3801 . . . 4 (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴)
31, 2sylib 208 . . 3 (𝜑 → (ℂ ∩ 𝐴) = 𝐴)
43raleqdv 3137 . 2 (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹𝑥) ↔ ∀𝑥𝐴 0 ≤ (𝐹𝑥)))
5 0cn 9992 . . . . . 6 0 ∈ ℂ
6 fnconstg 6060 . . . . . 6 (0 ∈ ℂ → (ℂ × {0}) Fn ℂ)
75, 6ax-mp 5 . . . . 5 (ℂ × {0}) Fn ℂ
8 df-0p 23377 . . . . . 6 0𝑝 = (ℂ × {0})
98fneq1i 5953 . . . . 5 (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ)
107, 9mpbir 221 . . . 4 0𝑝 Fn ℂ
1110a1i 11 . . 3 (𝜑 → 0𝑝 Fn ℂ)
12 0pledm.2 . . 3 (𝜑𝐹 Fn 𝐴)
13 cnex 9977 . . . 4 ℂ ∈ V
1413a1i 11 . . 3 (𝜑 → ℂ ∈ V)
15 ssexg 4774 . . . 4 ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V)
161, 13, 15sylancl 693 . . 3 (𝜑𝐴 ∈ V)
17 eqid 2621 . . 3 (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴)
18 0pval 23378 . . . 4 (𝑥 ∈ ℂ → (0𝑝𝑥) = 0)
1918adantl 482 . . 3 ((𝜑𝑥 ∈ ℂ) → (0𝑝𝑥) = 0)
20 eqidd 2622 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
2111, 12, 14, 16, 17, 19, 20ofrfval 6870 . 2 (𝜑 → (0𝑝𝑟𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹𝑥)))
22 fnconstg 6060 . . . . 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 3806 . . 3 (𝐴𝐴) = 𝐴
26 c0ex 9994 . . . . 5 0 ∈ V
2726fvconst2 6434 . . . 4 (𝑥𝐴 → ((𝐴 × {0})‘𝑥) = 0)
2827adantl 482 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {0})‘𝑥) = 0)
2924, 12, 16, 16, 25, 28, 20ofrfval 6870 . 2 (𝜑 → ((𝐴 × {0}) ∘𝑟𝐹 ↔ ∀𝑥𝐴 0 ≤ (𝐹𝑥)))
304, 21, 293bitr4d 300 1 (𝜑 → (0𝑝𝑟𝐹 ↔ (𝐴 × {0}) ∘𝑟𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  cin 3559  wss 3560  {csn 4155   class class class wbr 4623   × cxp 5082   Fn wfn 5852  cfv 5857  𝑟 cofr 6861  cc 9894  0cc0 9896  cle 10035  0𝑝c0p 23376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-cnex 9952  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-mulcl 9958  ax-i2m1 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ofr 6863  df-0p 23377
This theorem is referenced by:  xrge0f  23438  itg20  23444  itg2const  23447  i1fibl  23514  itgitg1  23515  ftc1anclem5  33160  ftc1anclem7  33162
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