Theorem 0pledm 25182

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 4215	. . . 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 11203	. . . . . 6 ⊢ 0 ∈ ℂ
6		fnconstg 6777	. . . . . 6 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ)
7	5, 6	ax-mp 5	. . . . 5 ⊢ (ℂ × {0}) Fn ℂ
8		df-0p 25179	. . . . . 6 ⊢ 0𝑝 = (ℂ × {0})
9	8	fneq1i 6644	. . . . 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 11188	. . . 4 ⊢ ℂ ∈ V
14	13	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
15		ssexg 5323	. . . 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 25180	. . . 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 7677	. 2 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥)))
22		fnconstg 6777	. . . . 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 4218	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
26		c0ex 11205	. . . . 5 ⊢ 0 ∈ V
27	26	fvconst2 7202	. . . 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 7677	. 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 3947   ⊆ wss 3948  {csn 4628   class class class wbr 5148   × cxp 5674   Fn wfn 6536  ‘cfv 6541   ∘_r cofr 7666  ℂcc 11105  0cc0 11107   ≤ cle 11246  0_𝑝c0p 25178

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 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-cnex 11163  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-mulcl 11169  ax-i2m1 11175

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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ofr 7668  df-0p 25179

This theorem is referenced by:  xrge0f  25241  itg20  25247  itg2const  25250  i1fibl  25317  itgitg1  25318  ftc1anclem5  36554  ftc1anclem7  36556