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Theorem 0pledm 23246
 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 3779 . . . 4 (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴)
31, 2sylib 207 . . 3 (𝜑 → (ℂ ∩ 𝐴) = 𝐴)
43raleqdv 3121 . 2 (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹𝑥) ↔ ∀𝑥𝐴 0 ≤ (𝐹𝑥)))
5 0cn 9911 . . . . . 6 0 ∈ ℂ
6 fnconstg 6006 . . . . . 6 (0 ∈ ℂ → (ℂ × {0}) Fn ℂ)
75, 6ax-mp 5 . . . . 5 (ℂ × {0}) Fn ℂ
8 df-0p 23243 . . . . . 6 0𝑝 = (ℂ × {0})
98fneq1i 5899 . . . . 5 (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ)
107, 9mpbir 220 . . . 4 0𝑝 Fn ℂ
1110a1i 11 . . 3 (𝜑 → 0𝑝 Fn ℂ)
12 0pledm.2 . . 3 (𝜑𝐹 Fn 𝐴)
13 cnex 9896 . . . 4 ℂ ∈ V
1413a1i 11 . . 3 (𝜑 → ℂ ∈ V)
15 ssexg 4732 . . . 4 ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V)
161, 13, 15sylancl 693 . . 3 (𝜑𝐴 ∈ V)
17 eqid 2610 . . 3 (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴)
18 0pval 23244 . . . 4 (𝑥 ∈ ℂ → (0𝑝𝑥) = 0)
1918adantl 481 . . 3 ((𝜑𝑥 ∈ ℂ) → (0𝑝𝑥) = 0)
20 eqidd 2611 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
2111, 12, 14, 16, 17, 19, 20ofrfval 6803 . 2 (𝜑 → (0𝑝𝑟𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹𝑥)))
22 fnconstg 6006 . . . . 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 3784 . . 3 (𝐴𝐴) = 𝐴
26 c0ex 9913 . . . . 5 0 ∈ V
2726fvconst2 6374 . . . 4 (𝑥𝐴 → ((𝐴 × {0})‘𝑥) = 0)
2827adantl 481 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {0})‘𝑥) = 0)
2924, 12, 16, 16, 25, 28, 20ofrfval 6803 . 2 (𝜑 → ((𝐴 × {0}) ∘𝑟𝐹 ↔ ∀𝑥𝐴 0 ≤ (𝐹𝑥)))
304, 21, 293bitr4d 299 1 (𝜑 → (0𝑝𝑟𝐹 ↔ (𝐴 × {0}) ∘𝑟𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {csn 4125   class class class wbr 4583   × cxp 5036   Fn wfn 5799  ‘cfv 5804   ∘𝑟 cofr 6794  ℂcc 9813  0cc0 9815   ≤ cle 9954  0𝑝c0p 23242 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-cnex 9871  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-i2m1 9883 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ofr 6796  df-0p 23243 This theorem is referenced by:  xrge0f  23304  itg20  23310  itg2const  23313  i1fibl  23380  itgitg1  23381  ftc1anclem5  32659  ftc1anclem7  32661
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