ip2di - Metamath Proof Explorer

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Theorem ip2di 21185

Description: Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (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.g	⊢ + = (+_g‘𝑊)
ipdir.p	⊢ ⨣ = (+_g‘𝐹)
ip2di.1	⊢ (𝜑 → 𝑊 ∈ PreHil)
ip2di.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ip2di.3	⊢ (𝜑 → 𝐵 ∈ 𝑉)
ip2di.4	⊢ (𝜑 → 𝐶 ∈ 𝑉)
ip2di.5	⊢ (𝜑 → 𝐷 ∈ 𝑉)

**Assertion**
Ref	Expression
ip2di	⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶))))

**Proof of Theorem ip2di**
Step	Hyp	Ref	Expression
1		ip2di.1	. . 3 ⊢ (𝜑 → 𝑊 ∈ PreHil)
2		ip2di.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		ip2di.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
4		phllmod 21174	. . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod)
6		ip2di.4	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
7		ip2di.5	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
8		phllmhm.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
9		ipdir.g	. . . . 5 ⊢ + = (+_g‘𝑊)
10	8, 9	lmodvacl 20478	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 + 𝐷) ∈ 𝑉)
11	5, 6, 7, 10	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ 𝑉)
12		phlsrng.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
13		phllmhm.h	. . . 4 ⊢ , = (·_𝑖‘𝑊)
14		ipdir.p	. . . 4 ⊢ ⨣ = (+_g‘𝐹)
15	12, 13, 8, 9, 14	ipdir 21183	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ (𝐶 + 𝐷) ∈ 𝑉)) → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷))))
16	1, 2, 3, 11, 15	syl13anc 1372	. 2 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷))))
17	12, 13, 8, 9, 14	ipdi 21184	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)))
18	1, 2, 6, 7, 17	syl13anc 1372	. . 3 ⊢ (𝜑 → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)))
19	12, 13, 8, 9, 14	ipdi 21184	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷)))
20	1, 3, 6, 7, 19	syl13anc 1372	. . . 4 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷)))
21	12	phlsrng 21175	. . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)
22		srngring 20452	. . . . . 6 ⊢ (𝐹 ∈ *-Ring → 𝐹 ∈ Ring)
23		ringcmn 20092	. . . . . 6 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ CMnd)
24	1, 21, 22, 23	4syl 19	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ CMnd)
25		eqid 2732	. . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	12, 13, 8, 25	ipcl 21177	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 , 𝐶) ∈ (Base‘𝐹))
27	1, 3, 6, 26	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (𝐵 , 𝐶) ∈ (Base‘𝐹))
28	12, 13, 8, 25	ipcl 21177	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐵 , 𝐷) ∈ (Base‘𝐹))
29	1, 3, 7, 28	syl3anc 1371	. . . . 5 ⊢ (𝜑 → (𝐵 , 𝐷) ∈ (Base‘𝐹))
30	25, 14	cmncom 19660	. . . . 5 ⊢ ((𝐹 ∈ CMnd ∧ (𝐵 , 𝐶) ∈ (Base‘𝐹) ∧ (𝐵 , 𝐷) ∈ (Base‘𝐹)) → ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶)))
31	24, 27, 29, 30	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶)))
32	20, 31	eqtrd 2772	. . 3 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶)))
33	18, 32	oveq12d 7423	. 2 ⊢ (𝜑 → ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷))) = (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))))
34	12, 13, 8, 25	ipcl 21177	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘𝐹))
35	1, 2, 6, 34	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝐴 , 𝐶) ∈ (Base‘𝐹))
36	12, 13, 8, 25	ipcl 21177	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐴 , 𝐷) ∈ (Base‘𝐹))
37	1, 2, 7, 36	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝐴 , 𝐷) ∈ (Base‘𝐹))
38	25, 14	cmn4 19663	. . 3 ⊢ ((𝐹 ∈ CMnd ∧ ((𝐴 , 𝐶) ∈ (Base‘𝐹) ∧ (𝐴 , 𝐷) ∈ (Base‘𝐹)) ∧ ((𝐵 , 𝐷) ∈ (Base‘𝐹) ∧ (𝐵 , 𝐶) ∈ (Base‘𝐹))) → (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶))))
39	24, 35, 37, 29, 27, 38	syl122anc 1379	. 2 ⊢ (𝜑 → (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶))))
40	16, 33, 39	3eqtrd 2776	1 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 +_gcplusg 17193 Scalarcsca 17196 ·_𝑖cip 17198 CMndccmn 19642 Ringcrg 20049 *-Ringcsr 20444 LModclmod 20463 PreHilcphl 21168

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-ghm 19084 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-rnghom 20243 df-staf 20445 df-srng 20446 df-lmod 20465 df-lmhm 20625 df-lvec 20706 df-sra 20777 df-rgmod 20778 df-phl 21170

This theorem is referenced by: cph2di 24715

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