Theorem 0pledm 25190

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	⊢ (𝜑 → 𝐴 ⊆ ℂ)
0pledm.2	⊢ (𝜑 → 𝐹 Fn 𝐴)

Assertion

Ref	Expression
0pledm	⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹))

Proof of Theorem 0pledm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0pledm.1	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
2		sseqin2 4216	. . . 4 ⊢ (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴)
3	1, 2	sylib 217	. . 3 ⊢ (𝜑 → (ℂ ∩ 𝐴) = 𝐴)
4	3	raleqdv 3326	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥)))
5		0cn 11206	. . . . . 6 ⊢ 0 ∈ ℂ
6		fnconstg 6780	. . . . . 6 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ)
7	5, 6	ax-mp 5	. . . . 5 ⊢ (ℂ × {0}) Fn ℂ
8		df-0p 25187	. . . . . 6 ⊢ 0𝑝 = (ℂ × {0})
9	8	fneq1i 6647	. . . . 5 ⊢ (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ)
10	7, 9	mpbir 230	. . . 4 ⊢ 0𝑝 Fn ℂ
11	10	a1i 11	. . 3 ⊢ (𝜑 → 0𝑝 Fn ℂ)
12		0pledm.2	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
13		cnex 11191	. . . 4 ⊢ ℂ ∈ V
14	13	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
15		ssexg 5324	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V)
16	1, 13, 15	sylancl 587	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
17		eqid 2733	. . 3 ⊢ (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴)
18		0pval 25188	. . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0)
19	18	adantl 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0)
20		eqidd 2734	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
21	11, 12, 14, 16, 17, 19, 20	ofrfval 7680	. 2 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥)))
22		fnconstg 6780	. . . . 5 ⊢ (0 ∈ ℂ → (𝐴 × {0}) Fn 𝐴)
23	5, 22	ax-mp 5	. . . 4 ⊢ (𝐴 × {0}) Fn 𝐴
24	23	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 × {0}) Fn 𝐴)
25		inidm 4219	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
26		c0ex 11208	. . . . 5 ⊢ 0 ∈ V
27	26	fvconst2 7205	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0)
28	27	adantl 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0)
29	24, 12, 16, 16, 25, 28, 20	ofrfval 7680	. 2 ⊢ (𝜑 → ((𝐴 × {0}) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥)))
30	4, 21, 29	3bitr4d 311	1 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949  {csn 4629   class class class wbr 5149   × cxp 5675   Fn wfn 6539  ‘cfv 6544   ∘_r cofr 7669  ℂcc 11108  0cc0 11110   ≤ cle 11249  0_𝑝c0p 25186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-cnex 11166  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-mulcl 11172  ax-i2m1 11178

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ofr 7671  df-0p 25187

This theorem is referenced by:  xrge0f  25249  itg20  25255  itg2const  25258  i1fibl  25325  itgitg1  25326  ftc1anclem5  36565  ftc1anclem7  36567