Theorem ipffval 22009

Description: The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)

Hypotheses

Ref	Expression
ipffval.1	⊢ 𝑉 = (Base‘𝑊)
ipffval.2	⊢ , = (·𝑖‘𝑊)
ipffval.3	⊢ · = (·if‘𝑊)

Assertion

Ref	Expression
ipffval	⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))

Distinct variable groups:   𝑥,𝑦, ,   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   · (𝑥, 𝑦)

Proof of Theorem ipffval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ipffval.3	. 2 ⊢ · = (·if‘𝑊)
2		fveq2 6916	. . . . . 6 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
3		ipffval.1	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
4	2, 3	eqtr4di 2816	. . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
5		fveq2 6916	. . . . . . 7 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = (·𝑖‘𝑊))
6		ipffval.2	. . . . . . 7 ⊢ , = (·𝑖‘𝑊)
7	5, 6	eqtr4di 2816	. . . . . 6 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = , )
8	7	oveqd 7468	. . . . 5 ⊢ (𝑔 = 𝑊 → (𝑥(·𝑖‘𝑔)𝑦) = (𝑥 , 𝑦))
9	4, 4, 8	mpoeq123dv 7526	. . . 4 ⊢ (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
10		df-ipf 21988	. . . 4 ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)))
11	3	fvexi 6930	. . . . 5 ⊢ 𝑉 ∈ V
12	6	fvexi 6930	. . . . . . 7 ⊢ , ∈ V
13	12	rnex 7956	. . . . . 6 ⊢ ran , ∈ V
14		p0ex 5353	. . . . . 6 ⊢ {∅} ∈ V
15	13, 14	unex 7793	. . . . 5 ⊢ (ran , ∪ {∅}) ∈ V
16		df-ov 7454	. . . . . . 7 ⊢ (𝑥 , 𝑦) = ( , ‘⟨𝑥, 𝑦⟩)
17		fvrn0 6944	. . . . . . 7 ⊢ ( , ‘⟨𝑥, 𝑦⟩) ∈ (ran , ∪ {∅})
18	16, 17	eqeltri 2860	. . . . . 6 ⊢ (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
19	18	rgen2w 3085	. . . . 5 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
20	11, 11, 15, 19	mpoexw 8126	. . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V
21	9, 10, 20	fvmpt 7024	. . 3 ⊢ (𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
22		fvprc 6908	. . . 4 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = ∅)
23		fvprc 6908	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
24	3, 23	eqtrid 2810	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
25	24	olcd 888	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑉 = ∅ ∨ 𝑉 = ∅))
26		0mpo0 7534	. . . . 5 ⊢ ((𝑉 = ∅ ∨ 𝑉 = ∅) → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = ∅)
27	25, 26	syl 18	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = ∅)
28	22, 27	eqtr4d 2801	. . 3 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
29	21, 28	pm2.61i 184	. 2 ⊢ (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
30	1, 29	eqtri 2786	1 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4840  ran crn 5659  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  Basecbs 17431  ·_𝑖cip 17477  ·_ifcipf 21986

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-ipf 21988

This theorem is used by:  ipfval  22010  ipfeq  22011  ipffn  22012  phlipf  22013  phssip  22019