Theorem 0pledm 25994

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 4169	. . . 4 ⊢ (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴)
3	1, 2	sylib 221	. . 3 ⊢ (𝜑 → (ℂ ∩ 𝐴) = 𝐴)
4	3	raleqdv 3320	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥)))
5		0cn 11298	. . . . . 6 ⊢ 0 ∈ ℂ
6		fnconstg 6770	. . . . . 6 ⊢ (0 ∈ ℂ → (ℂ × {0}) Fn ℂ)
7	5, 6	ax-mp 5	. . . . 5 ⊢ (ℂ × {0}) Fn ℂ
8		df-0p 25991	. . . . . 6 ⊢ 0𝑝 = (ℂ × {0})
9	8	fneq1i 6636	. . . . 5 ⊢ (0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn ℂ)
10	7, 9	mpbir 234	. . . 4 ⊢ 0𝑝 Fn ℂ
11	10	a1i 11	. . 3 ⊢ (𝜑 → 0𝑝 Fn ℂ)
12		0pledm.2	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
13		cnex 11281	. . . 4 ⊢ ℂ ∈ V
14	13	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
15		ssexg 5281	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V)
16	1, 13, 15	sylancl 598	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
17		eqid 2761	. . 3 ⊢ (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴)
18		0pval 25992	. . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0)
19	18	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0)
20		eqidd 2762	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
21	11, 12, 14, 16, 17, 19, 20	ofrfval 7703	. 2 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥)))
22		fnconstg 6770	. . . . 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 4172	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
26		c0ex 11300	. . . . 5 ⊢ 0 ∈ V
27	26	fvconst2 7210	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0)
28	27	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0)
29	24, 12, 16, 16, 25, 28, 20	ofrfval 7703	. 2 ⊢ (𝜑 → ((𝐴 × {0}) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥)))
30	4, 21, 29	3bitr4d 314	1 ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  {csn 4584   class class class wbr 5103   × cxp 5649   Fn wfn 6533  ‘cfv 6538   ∘_r cofr 7692  ℂcc 11198  0cc0 11200   ≤ cle 11344  0_𝑝c0p 25990

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-cnex 11256  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-mulcl 11262  ax-i2m1 11268

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ofr 7694  df-0p 25991

This theorem is used by:  xrge0f  26052  itg20  26058  itg2const  26061  i1fibl  26128  itgitg1  26129  ftc1anclem5  38615  ftc1anclem7  38617