Theorem ipffval 19812

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 6103	. . . . . 6 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
3		ipffval.1	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
4	2, 3	syl6eqr 2662	. . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
5		fveq2 6103	. . . . . . 7 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = (·𝑖‘𝑊))
6		ipffval.2	. . . . . . 7 ⊢ , = (·𝑖‘𝑊)
7	5, 6	syl6eqr 2662	. . . . . 6 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = , )
8	7	oveqd 6566	. . . . 5 ⊢ (𝑔 = 𝑊 → (𝑥(·𝑖‘𝑔)𝑦) = (𝑥 , 𝑦))
9	4, 4, 8	mpt2eq123dv 6615	. . . 4 ⊢ (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
10		df-ipf 19791	. . . 4 ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)))
11		df-ov 6552	. . . . . . . 8 ⊢ (𝑥 , 𝑦) = ( , ‘⟨𝑥, 𝑦⟩)
12		fvrn0 6126	. . . . . . . 8 ⊢ ( , ‘⟨𝑥, 𝑦⟩) ∈ (ran , ∪ {∅})
13	11, 12	eqeltri 2684	. . . . . . 7 ⊢ (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
14	13	rgen2w 2909	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
15		eqid 2610	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
16	15	fmpt2 7126	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅}) ↔ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅}))
17	14, 16	mpbi 219	. . . . 5 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅})
18		fvex 6113	. . . . . . 7 ⊢ (Base‘𝑊) ∈ V
19	3, 18	eqeltri 2684	. . . . . 6 ⊢ 𝑉 ∈ V
20	19, 19	xpex 6860	. . . . 5 ⊢ (𝑉 × 𝑉) ∈ V
21		fvex 6113	. . . . . . . 8 ⊢ (·𝑖‘𝑊) ∈ V
22	6, 21	eqeltri 2684	. . . . . . 7 ⊢ , ∈ V
23	22	rnex 6992	. . . . . 6 ⊢ ran , ∈ V
24		p0ex 4779	. . . . . 6 ⊢ {∅} ∈ V
25	23, 24	unex 6854	. . . . 5 ⊢ (ran , ∪ {∅}) ∈ V
26		fex2 7014	. . . . 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 1416	. . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V
28	9, 10, 27	fvmpt 6191	. . 3 ⊢ (𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
29		fvprc 6097	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = ∅)
30		mpt20 6623	. . . . 5 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)) = ∅
31	29, 30	syl6eqr 2662	. . . 4 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
32		fvprc 6097	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
33	3, 32	syl5eq 2656	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
34		mpt2eq12 6613	. . . . 5 ⊢ ((𝑉 = ∅ ∧ 𝑉 = ∅) → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
35	33, 33, 34	syl2anc 691	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
36	31, 35	eqtr4d 2647	. . 3 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
37	28, 36	pm2.61i 175	. 2 ⊢ (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
38	1, 37	eqtri 2632	1 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  ⟨cop 4131   × cxp 5036  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  ·_𝑖cip 15773  ·_ifcipf 19789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-ipf 19791

This theorem is referenced by:  ipfval  19813  ipfeq  19814  ipffn  19815  phlipf  19816  phssip  19822