Theorem 0pledm 25885

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 4176	. . . 4 ⊢ (𝐴 ⊆ ℂ ↔ (ℂ ∩ 𝐴) = 𝐴)
3	1, 2	sylib 221	. . 3 ⊢ (𝜑 → (ℂ ∩ 𝐴) = 𝐴)
4	3	raleqdv 3325	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (ℂ ∩ 𝐴)0 ≤ (𝐹‘𝑥) ↔ ∀𝑥 ∈ 𝐴 0 ≤ (𝐹‘𝑥)))
5		0cn 11215	. . . . . 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 25882	. . . . . 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 11198	. . . 4 ⊢ ℂ ∈ V
14	13	a1i 11	. . 3 ⊢ (𝜑 → ℂ ∈ V)
15		ssexg 5292	. . . 4 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ∈ V) → 𝐴 ∈ V)
16	1, 13, 15	sylancl 598	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
17		eqid 2765	. . 3 ⊢ (ℂ ∩ 𝐴) = (ℂ ∩ 𝐴)
18		0pval 25883	. . . 4 ⊢ (𝑥 ∈ ℂ → (0𝑝‘𝑥) = 0)
19	18	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0𝑝‘𝑥) = 0)
20		eqidd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
21	11, 12, 14, 16, 17, 19, 20	ofrfval 7694	. 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 4179	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
26		c0ex 11217	. . . . 5 ⊢ 0 ∈ V
27	26	fvconst2 7209	. . . 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 7694	. 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 2146  ∀wral 3081  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  {csn 4591   class class class wbr 5111   × cxp 5661   Fn wfn 6535  ‘cfv 6540   ∘_r cofr 7683  ℂcc 11115  0cc0 11117   ≤ cle 11261  0_𝑝c0p 25881

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-cnex 11173  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-mulcl 11179  ax-i2m1 11185

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ofr 7685  df-0p 25882

This theorem is used by:  xrge0f  25943  itg20  25949  itg2const  25952  i1fibl  26020  itgitg1  26021  ftc1anclem5  38407  ftc1anclem7  38409