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Theorem 0pledm 24201
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 4189 . . . 4 (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴)
31, 2sylib 219 . . 3 (𝜑 → (ℂ ∩ 𝐴) = 𝐴)
43raleqdv 3413 . 2 (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹𝑥) ↔ ∀𝑥𝐴 0 ≤ (𝐹𝑥)))
5 0cn 10621 . . . . . 6 0 ∈ ℂ
6 fnconstg 6560 . . . . . 6 (0 ∈ ℂ → (ℂ × {0}) Fn ℂ)
75, 6ax-mp 5 . . . . 5 (ℂ × {0}) Fn ℂ
8 df-0p 24198 . . . . . 6 0𝑝 = (ℂ × {0})
98fneq1i 6443 . . . . 5 (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ)
107, 9mpbir 232 . . . 4 0𝑝 Fn ℂ
1110a1i 11 . . 3 (𝜑 → 0𝑝 Fn ℂ)
12 0pledm.2 . . 3 (𝜑𝐹 Fn 𝐴)
13 cnex 10606 . . . 4 ℂ ∈ V
1413a1i 11 . . 3 (𝜑 → ℂ ∈ V)
15 ssexg 5218 . . . 4 ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V)
161, 13, 15sylancl 586 . . 3 (𝜑𝐴 ∈ V)
17 eqid 2818 . . 3 (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴)
18 0pval 24199 . . . 4 (𝑥 ∈ ℂ → (0𝑝𝑥) = 0)
1918adantl 482 . . 3 ((𝜑𝑥 ∈ ℂ) → (0𝑝𝑥) = 0)
20 eqidd 2819 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
2111, 12, 14, 16, 17, 19, 20ofrfval 7406 . 2 (𝜑 → (0𝑝r𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹𝑥)))
22 fnconstg 6560 . . . . 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 4192 . . 3 (𝐴𝐴) = 𝐴
26 c0ex 10623 . . . . 5 0 ∈ V
2726fvconst2 6958 . . . 4 (𝑥𝐴 → ((𝐴 × {0})‘𝑥) = 0)
2827adantl 482 . . 3 ((𝜑𝑥𝐴) → ((𝐴 × {0})‘𝑥) = 0)
2924, 12, 16, 16, 25, 28, 20ofrfval 7406 . 2 (𝜑 → ((𝐴 × {0}) ∘r𝐹 ↔ ∀𝑥𝐴 0 ≤ (𝐹𝑥)))
304, 21, 293bitr4d 312 1 (𝜑 → (0𝑝r𝐹 ↔ (𝐴 × {0}) ∘r𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cin 3932  wss 3933  {csn 4557   class class class wbr 5057   × cxp 5546   Fn wfn 6343  cfv 6348  r cofr 7397  cc 10523  0cc0 10525  cle 10664  0𝑝c0p 24197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-cnex 10581  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ofr 7399  df-0p 24198
This theorem is referenced by:  xrge0f  24259  itg20  24265  itg2const  24268  i1fibl  24335  itgitg1  24336  ftc1anclem5  34852  ftc1anclem7  34854
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