Theorem ipffval 20210

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 6333	. . . . . 6 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = (Base‘𝑊))
3		ipffval.1	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
4	2, 3	syl6eqr 2823	. . . . 5 ⊢ (𝑔 = 𝑊 → (Base‘𝑔) = 𝑉)
5		fveq2 6333	. . . . . . 7 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = (·𝑖‘𝑊))
6		ipffval.2	. . . . . . 7 ⊢ , = (·𝑖‘𝑊)
7	5, 6	syl6eqr 2823	. . . . . 6 ⊢ (𝑔 = 𝑊 → (·𝑖‘𝑔) = , )
8	7	oveqd 6813	. . . . 5 ⊢ (𝑔 = 𝑊 → (𝑥(·𝑖‘𝑔)𝑦) = (𝑥 , 𝑦))
9	4, 4, 8	mpt2eq123dv 6868	. . . 4 ⊢ (𝑔 = 𝑊 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
10		df-ipf 20189	. . . 4 ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)))
11		df-ov 6799	. . . . . . . 8 ⊢ (𝑥 , 𝑦) = ( , ‘⟨𝑥, 𝑦⟩)
12		fvrn0 6359	. . . . . . . 8 ⊢ ( , ‘⟨𝑥, 𝑦⟩) ∈ (ran , ∪ {∅})
13	11, 12	eqeltri 2846	. . . . . . 7 ⊢ (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
14	13	rgen2w 3074	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 , 𝑦) ∈ (ran , ∪ {∅})
15		eqid 2771	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
16	15	fmpt2 7391	. . . . . 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	3	fvexi 6345	. . . . . 6 ⊢ 𝑉 ∈ V
19	18, 18	xpex 7113	. . . . 5 ⊢ (𝑉 × 𝑉) ∈ V
20	6	fvexi 6345	. . . . . . 7 ⊢ , ∈ V
21	20	rnex 7251	. . . . . 6 ⊢ ran , ∈ V
22		p0ex 4985	. . . . . 6 ⊢ {∅} ∈ V
23	21, 22	unex 7107	. . . . 5 ⊢ (ran , ∪ {∅}) ∈ V
24		fex2 7272	. . . . 5 ⊢ (((𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)):(𝑉 × 𝑉)⟶(ran , ∪ {∅}) ∧ (𝑉 × 𝑉) ∈ V ∧ (ran , ∪ {∅}) ∈ V) → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V)
25	17, 19, 23, 24	mp3an 1572	. . . 4 ⊢ (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ V
26	9, 10, 25	fvmpt 6426	. . 3 ⊢ (𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
27		fvprc 6327	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = ∅)
28		mpt20 6876	. . . . 5 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)) = ∅
29	27, 28	syl6eqr 2823	. . . 4 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
30		fvprc 6327	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
31	3, 30	syl5eq 2817	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝑉 = ∅)
32		mpt2eq12 6866	. . . . 5 ⊢ ((𝑉 = ∅ ∧ 𝑉 = ∅) → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
33	31, 31, 32	syl2anc 573	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 , 𝑦)))
34	29, 33	eqtr4d 2808	. . 3 ⊢ (¬ 𝑊 ∈ V → (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦)))
35	26, 34	pm2.61i 176	. 2 ⊢ (·if‘𝑊) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
36	1, 35	eqtri 2793	1 ⊢ · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))

Syntax hints:  ¬ wn 3   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ∪ cun 3721  ∅c0 4063  {csn 4317  ⟨cop 4323   × cxp 5248  ran crn 5251  ⟶wf 6026  ‘cfv 6030  (class class class)co 6796   ↦ cmpt2 6798  Basecbs 16064  ·_𝑖cip 16154  ·_ifcipf 20187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-ipf 20189

This theorem is referenced by:  ipfval  20211  ipfeq  20212  ipffn  20213  phlipf  20214  phssip  20220