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Theorem ipassr 21698

Description: "Associative" law for second argument of inner product (compare ipass 21697). (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 486	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝑊 ∈ PreHil)
2		simpr3 1210	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐶 ∈ 𝐾)
3		simpr2 1209	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐵 ∈ 𝑉)
4		simpr1 1208	. . . . 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 21697	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐶 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) → ((𝐶 · 𝐵) , 𝐴) = (𝐶 × (𝐵 , 𝐴)))
12	1, 2, 3, 4, 11	syl13anc 1391	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝐶 · 𝐵) , 𝐴) = (𝐶 × (𝐵 , 𝐴)))
13	12	fveq2d 6871	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = ( ∗ ‘(𝐶 × (𝐵 , 𝐴))))
14		phllmod 21682	. . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
15	14	adantr 484	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝑊 ∈ LMod)
16	7, 5, 9, 8	lmodvscl 20945	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐶 · 𝐵) ∈ 𝑉)
17	15, 2, 3, 16	syl3anc 1390	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐶 · 𝐵) ∈ 𝑉)
18		ipassr.i	. . . . 5 ⊢ ∗ = (*_𝑟‘𝐹)
19	5, 6, 7, 18	ipcj 21686	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐶 · 𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = (𝐴 , (𝐶 · 𝐵)))
20	1, 17, 4, 19	syl3anc 1390	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = (𝐴 , (𝐶 · 𝐵)))
21	5	phlsrng 21683	. . . . 5 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)
22	21	adantr 484	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐹 ∈ *-Ring)
23	5, 6, 7, 8	ipcl 21685	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐵 , 𝐴) ∈ 𝐾)
24	1, 3, 4, 23	syl3anc 1390	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐵 , 𝐴) ∈ 𝐾)
25	18, 8, 10	srngmul 20901	. . . 4 ⊢ ((𝐹 ∈ *-Ring ∧ 𝐶 ∈ 𝐾 ∧ (𝐵 , 𝐴) ∈ 𝐾) → ( ∗ ‘(𝐶 × (𝐵 , 𝐴))) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
26	22, 2, 24, 25	syl3anc 1390	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘(𝐶 × (𝐵 , 𝐴))) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
27	13, 20, 26	3eqtr3d 2805	. 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (𝐶 · 𝐵)) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
28	5, 6, 7, 18	ipcj 21686	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ( ∗ ‘(𝐵 , 𝐴)) = (𝐴 , 𝐵))
29	1, 3, 4, 28	syl3anc 1390	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘(𝐵 , 𝐴)) = (𝐴 , 𝐵))
30	29	oveq1d 7411	. 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)) = ((𝐴 , 𝐵) × ( ∗ ‘𝐶)))
31	27, 30	eqtrd 2797	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 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 ._rcmulr 17287 *_𝑟cstv 17288 Scalarcsca 17289 ·_𝑠 cvsca 17290 ·_𝑖cip 17291 *-Ringcsr 20887 LModclmod 20927 PreHilcphl 21676

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-plusg 17299 df-mulr 17300 df-sca 17302 df-vsca 17303 df-ip 17304 df-0g 17470 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-mhm 18817 df-ghm 19254 df-mgp 20187 df-ur 20232 df-ring 20285 df-oppr 20386 df-rhm 20521 df-staf 20888 df-srng 20889 df-lmod 20929 df-lmhm 21089 df-lvec 21170 df-sra 21240 df-rgmod 21241 df-phl 21678

This theorem is referenced by: ipassr2 21699 cphassr 25274 tcphcphlem2 25298

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