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Theorem ipassr 21639

Description: "Associative" law for second argument of inner product (compare ipass 21638). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)

**Hypotheses**
Ref	Expression
phlsrng.f	⊢ 𝐹 = (Scalar‘𝑊)
phllmhm.h	⊢ , = (·_𝑖‘𝑊)
phllmhm.v	⊢ 𝑉 = (Base‘𝑊)
ipdir.f	⊢ 𝐾 = (Base‘𝐹)
ipass.s	⊢ · = ( ·_𝑠 ‘𝑊)
ipass.p	⊢ × = (._r‘𝐹)
ipassr.i	⊢ ∗ = (*_𝑟‘𝐹)

**Assertion**
Ref	Expression
ipassr	⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (𝐶 · 𝐵)) = ((𝐴 , 𝐵) × ( ∗ ‘𝐶)))

**Proof of Theorem ipassr**
Step	Hyp	Ref	Expression
1		simpl 482	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝑊 ∈ PreHil)
2		simpr3 1198	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐶 ∈ 𝐾)
3		simpr2 1197	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐵 ∈ 𝑉)
4		simpr1 1196	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐴 ∈ 𝑉)
5		phlsrng.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
6		phllmhm.h	. . . . . 6 ⊢ , = (·_𝑖‘𝑊)
7		phllmhm.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
8		ipdir.f	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
9		ipass.s	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑊)
10		ipass.p	. . . . . 6 ⊢ × = (._r‘𝐹)
11	5, 6, 7, 8, 9, 10	ipass 21638	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐶 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) → ((𝐶 · 𝐵) , 𝐴) = (𝐶 × (𝐵 , 𝐴)))
12	1, 2, 3, 4, 11	syl13anc 1375	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝐶 · 𝐵) , 𝐴) = (𝐶 × (𝐵 , 𝐴)))
13	12	fveq2d 6839	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = ( ∗ ‘(𝐶 × (𝐵 , 𝐴))))
14		phllmod 21623	. . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
15	14	adantr 480	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝑊 ∈ LMod)
16	7, 5, 9, 8	lmodvscl 20867	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐶 · 𝐵) ∈ 𝑉)
17	15, 2, 3, 16	syl3anc 1374	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐶 · 𝐵) ∈ 𝑉)
18		ipassr.i	. . . . 5 ⊢ ∗ = (*_𝑟‘𝐹)
19	5, 6, 7, 18	ipcj 21627	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐶 · 𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = (𝐴 , (𝐶 · 𝐵)))
20	1, 17, 4, 19	syl3anc 1374	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = (𝐴 , (𝐶 · 𝐵)))
21	5	phlsrng 21624	. . . . 5 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)
22	21	adantr 480	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐹 ∈ *-Ring)
23	5, 6, 7, 8	ipcl 21626	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐵 , 𝐴) ∈ 𝐾)
24	1, 3, 4, 23	syl3anc 1374	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐵 , 𝐴) ∈ 𝐾)
25	18, 8, 10	srngmul 20823	. . . 4 ⊢ ((𝐹 ∈ *-Ring ∧ 𝐶 ∈ 𝐾 ∧ (𝐵 , 𝐴) ∈ 𝐾) → ( ∗ ‘(𝐶 × (𝐵 , 𝐴))) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
26	22, 2, 24, 25	syl3anc 1374	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘(𝐶 × (𝐵 , 𝐴))) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
27	13, 20, 26	3eqtr3d 2780	. 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (𝐶 · 𝐵)) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
28	5, 6, 7, 18	ipcj 21627	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ( ∗ ‘(𝐵 , 𝐴)) = (𝐴 , 𝐵))
29	1, 3, 4, 28	syl3anc 1374	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘(𝐵 , 𝐴)) = (𝐴 , 𝐵))
30	29	oveq1d 7376	. 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)) = ((𝐴 , 𝐵) × ( ∗ ‘𝐶)))
31	27, 30	eqtrd 2772	1 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (𝐶 · 𝐵)) = ((𝐴 , 𝐵) × ( ∗ ‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6493 (class class class)co 7361 Basecbs 17173 ._rcmulr 17215 *_𝑟cstv 17216 Scalarcsca 17217 ·_𝑠 cvsca 17218 ·_𝑖cip 17219 *-Ringcsr 20809 LModclmod 20849 PreHilcphl 21617

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-ip 17232 df-0g 17398 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-mhm 18745 df-ghm 19182 df-mgp 20116 df-ur 20157 df-ring 20210 df-oppr 20311 df-rhm 20446 df-staf 20810 df-srng 20811 df-lmod 20851 df-lmhm 21012 df-lvec 21093 df-sra 21163 df-rgmod 21164 df-phl 21619

This theorem is referenced by: ipassr2 21640 cphassr 25192 tcphcphlem2 25216

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