Theorem 0pledm 25422

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 4214	. . . 4 ⊢ (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴)
3	1, 2	sylib 217	. . 3 ⊢ (𝜑 → (ℂ ∩ 𝐴) = 𝐴)
4	3	raleqdv 3323	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥)))
5		0cn 11210	. . . . . 6 ⊢ 0 ∈ ℂ
6		fnconstg 6778	. . . . . 6 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ)
7	5, 6	ax-mp 5	. . . . 5 ⊢ (ℂ × {0}) Fn ℂ
8		df-0p 25419	. . . . . 6 ⊢ 0𝑝 = (ℂ × {0})
9	8	fneq1i 6645	. . . . 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 11193	. . . 4 ⊢ ℂ ∈ V
14	13	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
15		ssexg 5322	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V)
16	1, 13, 15	sylancl 584	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
17		eqid 2730	. . 3 ⊢ (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴)
18		0pval 25420	. . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0)
19	18	adantl 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0)
20		eqidd 2731	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
21	11, 12, 14, 16, 17, 19, 20	ofrfval 7682	. 2 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥)))
22		fnconstg 6778	. . . . 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 4217	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
26		c0ex 11212	. . . . 5 ⊢ 0 ∈ V
27	26	fvconst2 7206	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0)
28	27	adantl 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0)
29	24, 12, 16, 16, 25, 28, 20	ofrfval 7682	. 2 ⊢ (𝜑 → ((𝐴 × {0}) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥)))
30	4, 21, 29	3bitr4d 310	1 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  Vcvv 3472   ∩ cin 3946   ⊆ wss 3947  {csn 4627   class class class wbr 5147   × cxp 5673   Fn wfn 6537  ‘cfv 6542   ∘_r cofr 7671  ℂcc 11110  0cc0 11112   ≤ cle 11253  0_𝑝c0p 25418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-cnex 11168  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-mulcl 11174  ax-i2m1 11180

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ofr 7673  df-0p 25419

This theorem is referenced by:  xrge0f  25481  itg20  25487  itg2const  25490  i1fibl  25557  itgitg1  25558  ftc1anclem5  36868  ftc1anclem7  36870