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Theorem 0pledm 23160
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 3775 . . . 4 (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴)
31, 2sylib 206 . . 3 (𝜑 → (ℂ ∩ 𝐴) = 𝐴)
43raleqdv 3117 . 2 (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹𝑥) ↔ ∀𝑥𝐴 0 ≤ (𝐹𝑥)))
5 0cn 9885 . . . . . 6 0 ∈ ℂ
6 fnconstg 5988 . . . . . 6 (0 ∈ ℂ → (ℂ × {0}) Fn ℂ)
75, 6ax-mp 5 . . . . 5 (ℂ × {0}) Fn ℂ
8 df-0p 23157 . . . . . 6 0𝑝 = (ℂ × {0})
98fneq1i 5882 . . . . 5 (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ)
107, 9mpbir 219 . . . 4 0𝑝 Fn ℂ
1110a1i 11 . . 3 (𝜑 → 0𝑝 Fn ℂ)
12 0pledm.2 . . 3 (𝜑𝐹 Fn 𝐴)
13 cnex 9870 . . . 4 ℂ ∈ V
1413a1i 11 . . 3 (𝜑 → ℂ ∈ V)
15 ssexg 4724 . . . 4 ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V)
161, 13, 15sylancl 692 . . 3 (𝜑𝐴 ∈ V)
17 eqid 2606 . . 3 (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴)
18 0pval 23158 . . . 4 (𝑥 ∈ ℂ → (0𝑝𝑥) = 0)
1918adantl 480 . . 3 ((𝜑𝑥 ∈ ℂ) → (0𝑝𝑥) = 0)
20 eqidd 2607 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
2111, 12, 14, 16, 17, 19, 20ofrfval 6777 . 2 (𝜑 → (0𝑝𝑟𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹𝑥)))
22 fnconstg 5988 . . . . 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 3780 . . 3 (𝐴𝐴) = 𝐴
26 c0ex 9887 . . . . 5 0 ∈ V
2726fvconst2 6349 . . . 4 (𝑥𝐴 → ((𝐴 × {0})‘𝑥) = 0)
2827adantl 480 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {0})‘𝑥) = 0)
2924, 12, 16, 16, 25, 28, 20ofrfval 6777 . 2 (𝜑 → ((𝐴 × {0}) ∘𝑟𝐹 ↔ ∀𝑥𝐴 0 ≤ (𝐹𝑥)))
304, 21, 293bitr4d 298 1 (𝜑 → (0𝑝𝑟𝐹 ↔ (𝐴 × {0}) ∘𝑟𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2892  Vcvv 3169  cin 3535  wss 3536  {csn 4121   class class class wbr 4574   × cxp 5023   Fn wfn 5782  cfv 5787  𝑟 cofr 6768  cc 9787  0cc0 9789  cle 9928  0𝑝c0p 23156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pr 4825  ax-cnex 9845  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-mulcl 9851  ax-i2m1 9857
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ofr 6770  df-0p 23157
This theorem is referenced by:  xrge0f  23218  itg20  23224  itg2const  23227  i1fibl  23294  itgitg1  23295  ftc1anclem5  32459  ftc1anclem7  32461
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