Theorem ipffval 21623

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 6827	. . . . . 6 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
3		ipffval.1	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
4	2, 3	eqtr4di 2792	. . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
5		fveq2 6827	. . . . . . 7 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = (·𝑖‘𝑊))
6		ipffval.2	. . . . . . 7 ⊢ , = (·𝑖‘𝑊)
7	5, 6	eqtr4di 2792	. . . . . 6 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = , )
8	7	oveqd 7373	. . . . 5 ⊢ (𝑔 = 𝑊 → (𝑥(·𝑖‘𝑔)𝑦) = (𝑥 , 𝑦))
9	4, 4, 8	mpoeq123dv 7431	. . . 4 ⊢ (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
10		df-ipf 21602	. . . 4 ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)))
11	3	fvexi 6841	. . . . 5 ⊢ 𝑉 ∈ V
12	6	fvexi 6841	. . . . . . 7 ⊢ , ∈ V
13	12	rnex 7850	. . . . . 6 ⊢ ran , ∈ V
14		p0ex 5313	. . . . . 6 ⊢ {∅} ∈ V
15	13, 14	unex 7687	. . . . 5 ⊢ (ran , ∪ {∅}) ∈ V
16		df-ov 7359	. . . . . . 7 ⊢ (𝑥 , 𝑦) = ( , ‘⟨𝑥, 𝑦⟩)
17		fvrn0 6855	. . . . . . 7 ⊢ ( , ‘⟨𝑥, 𝑦⟩) ∈ (ran , ∪ {∅})
18	16, 17	eqeltri 2835	. . . . . 6 ⊢ (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
19	18	rgen2w 3058	. . . . 5 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
20	11, 11, 15, 19	mpoexw 8020	. . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V
21	9, 10, 20	fvmpt 6935	. . 3 ⊢ (𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
22		fvprc 6819	. . . 4 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = ∅)
23		fvprc 6819	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
24	3, 23	eqtrid 2786	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
25	24	olcd 880	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑉 = ∅ ∨ 𝑉 = ∅))
26		0mpo0 7439	. . . . 5 ⊢ ((𝑉 = ∅ ∨ 𝑉 = ∅) → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = ∅)
27	25, 26	syl 17	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = ∅)
28	22, 27	eqtr4d 2777	. . 3 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
29	21, 28	pm2.61i 183	. 2 ⊢ (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
30	1, 29	eqtri 2762	1 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 853   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881  ∅c0 4261  {csn 4555  ⟨cop 4561  ran crn 5619  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  Basecbs 17170  ·_𝑖cip 17216  ·_ifcipf 21600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-ipf 21602

This theorem is referenced by:  ipfval  21624  ipfeq  21625  ipffn  21626  phlipf  21627  phssip  21633