ip2subdi - Metamath Proof Explorer

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Theorem ip2subdi 21196

Description: Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.)

**Hypotheses**
Ref	Expression
phlsrng.f	⊢ 𝐹 = (Scalar‘𝑊)
phllmhm.h	⊢ , = (·_𝑖‘𝑊)
phllmhm.v	⊢ 𝑉 = (Base‘𝑊)
ipsubdir.m	⊢ − = (-_g‘𝑊)
ipsubdir.s	⊢ 𝑆 = (-_g‘𝐹)
ip2subdi.p	⊢ + = (+_g‘𝐹)
ip2subdi.1	⊢ (𝜑 → 𝑊 ∈ PreHil)
ip2subdi.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ip2subdi.3	⊢ (𝜑 → 𝐵 ∈ 𝑉)
ip2subdi.4	⊢ (𝜑 → 𝐶 ∈ 𝑉)
ip2subdi.5	⊢ (𝜑 → 𝐷 ∈ 𝑉)

**Assertion**
Ref	Expression
ip2subdi	⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷))𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))))

**Proof of Theorem ip2subdi**
Step	Hyp	Ref	Expression
1		eqid 2732	. . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		ip2subdi.p	. . . 4 ⊢ + = (+_g‘𝐹)
3		ipsubdir.s	. . . 4 ⊢ 𝑆 = (-_g‘𝐹)
4		ip2subdi.1	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ PreHil)
5		phllmod 21182	. . . . . . 7 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod)
7		phlsrng.f	. . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑊)
8	7	lmodring 20478	. . . . . 6 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
9	6, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ Ring)
10		ringabl 20097	. . . . 5 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Abel)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Abel)
12		ip2subdi.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
13		ip2subdi.4	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
14		phllmhm.h	. . . . . 6 ⊢ , = (·_𝑖‘𝑊)
15		phllmhm.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
16	7, 14, 15, 1	ipcl 21185	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘𝐹))
17	4, 12, 13, 16	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐴 , 𝐶) ∈ (Base‘𝐹))
18		ip2subdi.5	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
19	7, 14, 15, 1	ipcl 21185	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐴 , 𝐷) ∈ (Base‘𝐹))
20	4, 12, 18, 19	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐴 , 𝐷) ∈ (Base‘𝐹))
21		ip2subdi.3	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
22	7, 14, 15, 1	ipcl 21185	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 , 𝐶) ∈ (Base‘𝐹))
23	4, 21, 13, 22	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐵 , 𝐶) ∈ (Base‘𝐹))
24	1, 2, 3, 11, 17, 20, 23	ablsubsub4 19685	. . 3 ⊢ (𝜑 → (((𝐴 , 𝐶)𝑆(𝐴 , 𝐷))𝑆(𝐵 , 𝐶)) = ((𝐴 , 𝐶)𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))))
25	24	oveq1d 7423	. 2 ⊢ (𝜑 → ((((𝐴 , 𝐶)𝑆(𝐴 , 𝐷))𝑆(𝐵 , 𝐶)) + (𝐵 , 𝐷)) = (((𝐴 , 𝐶)𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))) + (𝐵 , 𝐷)))
26		ipsubdir.m	. . . . . 6 ⊢ − = (-_g‘𝑊)
27	15, 26	lmodvsubcl 20516	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 − 𝐷) ∈ 𝑉)
28	6, 13, 18, 27	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐷) ∈ 𝑉)
29	7, 14, 15, 26, 3	ipsubdir 21194	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ (𝐶 − 𝐷) ∈ 𝑉)) → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = ((𝐴 , (𝐶 − 𝐷))𝑆(𝐵 , (𝐶 − 𝐷))))
30	4, 12, 21, 28, 29	syl13anc 1372	. . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = ((𝐴 , (𝐶 − 𝐷))𝑆(𝐵 , (𝐶 − 𝐷))))
31	7, 14, 15, 26, 3	ipsubdi 21195	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 , (𝐶 − 𝐷)) = ((𝐴 , 𝐶)𝑆(𝐴 , 𝐷)))
32	4, 12, 13, 18, 31	syl13anc 1372	. . . 4 ⊢ (𝜑 → (𝐴 , (𝐶 − 𝐷)) = ((𝐴 , 𝐶)𝑆(𝐴 , 𝐷)))
33	7, 14, 15, 26, 3	ipsubdi 21195	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 , (𝐶 − 𝐷)) = ((𝐵 , 𝐶)𝑆(𝐵 , 𝐷)))
34	4, 21, 13, 18, 33	syl13anc 1372	. . . 4 ⊢ (𝜑 → (𝐵 , (𝐶 − 𝐷)) = ((𝐵 , 𝐶)𝑆(𝐵 , 𝐷)))
35	32, 34	oveq12d 7426	. . 3 ⊢ (𝜑 → ((𝐴 , (𝐶 − 𝐷))𝑆(𝐵 , (𝐶 − 𝐷))) = (((𝐴 , 𝐶)𝑆(𝐴 , 𝐷))𝑆((𝐵 , 𝐶)𝑆(𝐵 , 𝐷))))
36		ringgrp 20060	. . . . . 6 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp)
37	9, 36	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ Grp)
38	1, 3	grpsubcl 18902	. . . . 5 ⊢ ((𝐹 ∈ Grp ∧ (𝐴 , 𝐶) ∈ (Base‘𝐹) ∧ (𝐴 , 𝐷) ∈ (Base‘𝐹)) → ((𝐴 , 𝐶)𝑆(𝐴 , 𝐷)) ∈ (Base‘𝐹))
39	37, 17, 20, 38	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((𝐴 , 𝐶)𝑆(𝐴 , 𝐷)) ∈ (Base‘𝐹))
40	7, 14, 15, 1	ipcl 21185	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐵 , 𝐷) ∈ (Base‘𝐹))
41	4, 21, 18, 40	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐵 , 𝐷) ∈ (Base‘𝐹))
42	1, 2, 3, 11, 39, 23, 41	ablsubsub 19684	. . 3 ⊢ (𝜑 → (((𝐴 , 𝐶)𝑆(𝐴 , 𝐷))𝑆((𝐵 , 𝐶)𝑆(𝐵 , 𝐷))) = ((((𝐴 , 𝐶)𝑆(𝐴 , 𝐷))𝑆(𝐵 , 𝐶)) + (𝐵 , 𝐷)))
43	30, 35, 42	3eqtrd 2776	. 2 ⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = ((((𝐴 , 𝐶)𝑆(𝐴 , 𝐷))𝑆(𝐵 , 𝐶)) + (𝐵 , 𝐷)))
44	1, 2	ringacl 20094	. . . 4 ⊢ ((𝐹 ∈ Ring ∧ (𝐴 , 𝐷) ∈ (Base‘𝐹) ∧ (𝐵 , 𝐶) ∈ (Base‘𝐹)) → ((𝐴 , 𝐷) + (𝐵 , 𝐶)) ∈ (Base‘𝐹))
45	9, 20, 23, 44	syl3anc 1371	. . 3 ⊢ (𝜑 → ((𝐴 , 𝐷) + (𝐵 , 𝐶)) ∈ (Base‘𝐹))
46	1, 2, 3	abladdsub 19679	. . 3 ⊢ ((𝐹 ∈ Abel ∧ ((𝐴 , 𝐶) ∈ (Base‘𝐹) ∧ (𝐵 , 𝐷) ∈ (Base‘𝐹) ∧ ((𝐴 , 𝐷) + (𝐵 , 𝐶)) ∈ (Base‘𝐹))) → (((𝐴 , 𝐶) + (𝐵 , 𝐷))𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))) = (((𝐴 , 𝐶)𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))) + (𝐵 , 𝐷)))
47	11, 17, 41, 45, 46	syl13anc 1372	. 2 ⊢ (𝜑 → (((𝐴 , 𝐶) + (𝐵 , 𝐷))𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))) = (((𝐴 , 𝐶)𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))) + (𝐵 , 𝐷)))
48	25, 43, 47	3eqtr4d 2782	1 ⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷))𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 +_gcplusg 17196 Scalarcsca 17199 ·_𝑖cip 17201 Grpcgrp 18818 -_gcsg 18820 Abelcabl 19648 Ringcrg 20055 LModclmod 20470 PreHilcphl 21176

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-grp 18821 df-minusg 18822 df-sbg 18823 df-ghm 19089 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-oppr 20149 df-rnghom 20250 df-staf 20452 df-srng 20453 df-lmod 20472 df-lmhm 20632 df-lvec 20713 df-sra 20784 df-rgmod 20785 df-phl 21178

This theorem is referenced by: cph2subdi 24726 ipcau2 24750 tcphcphlem1 24751

W3C validator