Theorem ipffval 21861

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 6878	. . . . . 6 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
3		ipffval.1	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
4	2, 3	eqtr4di 2813	. . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
5		fveq2 6878	. . . . . . 7 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = (·𝑖‘𝑊))
6		ipffval.2	. . . . . . 7 ⊢ , = (·𝑖‘𝑊)
7	5, 6	eqtr4di 2813	. . . . . 6 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = , )
8	7	oveqd 7430	. . . . 5 ⊢ (𝑔 = 𝑊 → (𝑥(·𝑖‘𝑔)𝑦) = (𝑥 , 𝑦))
9	4, 4, 8	mpoeq123dv 7488	. . . 4 ⊢ (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
10		df-ipf 21840	. . . 4 ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)))
11	3	fvexi 6892	. . . . 5 ⊢ 𝑉 ∈ V
12	6	fvexi 6892	. . . . . . 7 ⊢ , ∈ V
13	12	rnex 7907	. . . . . 6 ⊢ ran , ∈ V
14		p0ex 5349	. . . . . 6 ⊢ {∅} ∈ V
15	13, 14	unex 7746	. . . . 5 ⊢ (ran , ∪ {∅}) ∈ V
16		df-ov 7416	. . . . . . 7 ⊢ (𝑥 , 𝑦) = ( , ‘⟨𝑥, 𝑦⟩)
17		fvrn0 6906	. . . . . . 7 ⊢ ( , ‘⟨𝑥, 𝑦⟩) ∈ (ran , ∪ {∅})
18	16, 17	eqeltri 2856	. . . . . 6 ⊢ (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
19	18	rgen2w 3081	. . . . 5 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
20	11, 11, 15, 19	mpoexw 8077	. . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V
21	9, 10, 20	fvmpt 6986	. . 3 ⊢ (𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
22		fvprc 6870	. . . 4 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = ∅)
23		fvprc 6870	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
24	3, 23	eqtrid 2807	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
25	24	olcd 888	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑉 = ∅ ∨ 𝑉 = ∅))
26		0mpo0 7496	. . . . 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 2798	. . 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 2783	1 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∪ cun 3897  ∅c0 4279  {csn 4584  ⟨cop 4590  ran crn 5656  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  Basecbs 17301  ·_𝑖cip 17347  ·_ifcipf 21838

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 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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-pw 4559  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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-ipf 21840

This theorem is used by:  ipfval  21862  ipfeq  21863  ipffn  21864  phlipf  21865  phssip  21871