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Theorem ipassr 20784

Description: "Associative" law for second argument of inner product (compare ipass 20783). (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 485	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝑊 ∈ PreHil)
2		simpr3 1192	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐶 ∈ 𝐾)
3		simpr2 1191	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐵 ∈ 𝑉)
4		simpr1 1190	. . . . 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 20783	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐶 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) → ((𝐶 · 𝐵) , 𝐴) = (𝐶 × (𝐵 , 𝐴)))
12	1, 2, 3, 4, 11	syl13anc 1368	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝐶 · 𝐵) , 𝐴) = (𝐶 × (𝐵 , 𝐴)))
13	12	fveq2d 6668	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = ( ∗ ‘(𝐶 × (𝐵 , 𝐴))))
14		phllmod 20768	. . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
15	14	adantr 483	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝑊 ∈ LMod)
16	7, 5, 9, 8	lmodvscl 19645	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐶 · 𝐵) ∈ 𝑉)
17	15, 2, 3, 16	syl3anc 1367	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐶 · 𝐵) ∈ 𝑉)
18		ipassr.i	. . . . 5 ⊢ ∗ = (*_𝑟‘𝐹)
19	5, 6, 7, 18	ipcj 20772	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐶 · 𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = (𝐴 , (𝐶 · 𝐵)))
20	1, 17, 4, 19	syl3anc 1367	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = (𝐴 , (𝐶 · 𝐵)))
21	5	phlsrng 20769	. . . . 5 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)
22	21	adantr 483	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐹 ∈ *-Ring)
23	5, 6, 7, 8	ipcl 20771	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐵 , 𝐴) ∈ 𝐾)
24	1, 3, 4, 23	syl3anc 1367	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐵 , 𝐴) ∈ 𝐾)
25	18, 8, 10	srngmul 19623	. . . 4 ⊢ ((𝐹 ∈ *-Ring ∧ 𝐶 ∈ 𝐾 ∧ (𝐵 , 𝐴) ∈ 𝐾) → ( ∗ ‘(𝐶 × (𝐵 , 𝐴))) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
26	22, 2, 24, 25	syl3anc 1367	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘(𝐶 × (𝐵 , 𝐴))) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
27	13, 20, 26	3eqtr3d 2864	. 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (𝐶 · 𝐵)) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
28	5, 6, 7, 18	ipcj 20772	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ( ∗ ‘(𝐵 , 𝐴)) = (𝐴 , 𝐵))
29	1, 3, 4, 28	syl3anc 1367	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘(𝐵 , 𝐴)) = (𝐴 , 𝐵))
30	29	oveq1d 7165	. 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)) = ((𝐴 , 𝐵) × ( ∗ ‘𝐶)))
31	27, 30	eqtrd 2856	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 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 ._rcmulr 16560 *_𝑟cstv 16561 Scalarcsca 16562 ·_𝑠 cvsca 16563 ·_𝑖cip 16564 *-Ringcsr 19609 LModclmod 19628 PreHilcphl 20762

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-ghm 18350 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-rnghom 19461 df-staf 19610 df-srng 19611 df-lmod 19630 df-lmhm 19788 df-lvec 19869 df-sra 19938 df-rgmod 19939 df-phl 20764

This theorem is referenced by: ipassr2 20785 cphassr 23810 tcphcphlem2 23833

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