Theorem ipffval 19993

Description: The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)

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 6191	. . . . . 6 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
3		ipffval.1	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
4	2, 3	syl6eqr 2674	. . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
5		fveq2 6191	. . . . . . 7 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = (·𝑖‘𝑊))
6		ipffval.2	. . . . . . 7 ⊢ , = (·𝑖‘𝑊)
7	5, 6	syl6eqr 2674	. . . . . 6 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = , )
8	7	oveqd 6667	. . . . 5 ⊢ (𝑔 = 𝑊 → (𝑥(·𝑖‘𝑔)𝑦) = (𝑥 , 𝑦))
9	4, 4, 8	mpt2eq123dv 6717	. . . 4 ⊢ (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
10		df-ipf 19972	. . . 4 ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)))
11		df-ov 6653	. . . . . . . 8 ⊢ (𝑥 , 𝑦) = ( , ‘⟨𝑥, 𝑦⟩)
12		fvrn0 6216	. . . . . . . 8 ⊢ ( , ‘⟨𝑥, 𝑦⟩) ∈ (ran , ∪ {∅})
13	11, 12	eqeltri 2697	. . . . . . 7 ⊢ (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
14	13	rgen2w 2925	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
15		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
16	15	fmpt2 7237	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅}) ↔ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅}))
17	14, 16	mpbi 220	. . . . 5 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅})
18		fvex 6201	. . . . . . 7 ⊢ (Base‘𝑊) ∈ V
19	3, 18	eqeltri 2697	. . . . . 6 ⊢ 𝑉 ∈ V
20	19, 19	xpex 6962	. . . . 5 ⊢ (𝑉 × 𝑉) ∈ V
21		fvex 6201	. . . . . . . 8 ⊢ (·𝑖‘𝑊) ∈ V
22	6, 21	eqeltri 2697	. . . . . . 7 ⊢ , ∈ V
23	22	rnex 7100	. . . . . 6 ⊢ ran , ∈ V
24		p0ex 4853	. . . . . 6 ⊢ {∅} ∈ V
25	23, 24	unex 6956	. . . . 5 ⊢ (ran , ∪ {∅}) ∈ V
26		fex2 7121	. . . . 5 ⊢ (((𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅}) ∧ (𝑉 × 𝑉) ∈ V ∧ (ran , ∪ {∅}) ∈ V) → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V)
27	17, 20, 25, 26	mp3an 1424	. . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V
28	9, 10, 27	fvmpt 6282	. . 3 ⊢ (𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
29		fvprc 6185	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = ∅)
30		mpt20 6725	. . . . 5 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)) = ∅
31	29, 30	syl6eqr 2674	. . . 4 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
32		fvprc 6185	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
33	3, 32	syl5eq 2668	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
34		mpt2eq12 6715	. . . . 5 ⊢ ((𝑉 = ∅ ∧ 𝑉 = ∅) → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
35	33, 33, 34	syl2anc 693	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
36	31, 35	eqtr4d 2659	. . 3 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
37	28, 36	pm2.61i 176	. 2 ⊢ (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
38	1, 37	eqtri 2644	1 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∪ cun 3572  ∅c0 3915  {csn 4177  ⟨cop 4183   × cxp 5112  ran crn 5115  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  ·_𝑖cip 15946  ·_ifcipf 19970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-ipf 19972

This theorem is referenced by:  ipfval  19994  ipfeq  19995  ipffn  19996  phlipf  19997  phssip  20003