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Theorem ipassi 30927

Description: Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ip1i.1	⊢ 𝑋 = (BaseSet‘𝑈)
ip1i.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
ip1i.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
ip1i.7	⊢ 𝑃 = (·_𝑖OLD‘𝑈)
ip1i.9	⊢ 𝑈 ∈ CPreHil_OLD

**Assertion**
Ref	Expression
ipassi	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

**Proof of Theorem ipassi**
Step	Hyp	Ref	Expression
1		oveq2 7368	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐴𝑆𝐵) = (𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))))
2	1	oveq1d 7375	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → ((𝐴𝑆𝐵)𝑃𝐶) = ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃𝐶))
3		oveq1 7367	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐵𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶))
4	3	oveq2d 7376	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐴 · (𝐵𝑃𝐶)) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶)))
5	2, 4	eqeq12d 2753	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)) ↔ ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃𝐶) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶))))
6	5	imbi2d 340	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → ((𝐴 ∈ ℂ → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) ↔ (𝐴 ∈ ℂ → ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃𝐶) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶)))))
7		oveq2 7368	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)) → ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃𝐶) = ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈))))
8		oveq2 7368	. . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)) → (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈))))
9	8	oveq2d 7376	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)) → (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶)) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)))))
10	7, 9	eqeq12d 2753	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)) → (((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃𝐶) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶)) ↔ ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈))) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈))))))
11	10	imbi2d 340	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)) → ((𝐴 ∈ ℂ → ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃𝐶) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶))) ↔ (𝐴 ∈ ℂ → ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈))) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)))))))
12		ip1i.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
13		ip1i.2	. . . . 5 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
14		ip1i.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
15		ip1i.7	. . . . 5 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
16		ip1i.9	. . . . 5 ⊢ 𝑈 ∈ CPreHil_OLD
17		eqid 2737	. . . . . 6 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
18	12, 17, 16	elimph 30906	. . . . 5 ⊢ if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) ∈ 𝑋
19	12, 17, 16	elimph 30906	. . . . 5 ⊢ if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)) ∈ 𝑋
20	12, 13, 14, 15, 16, 18, 19	ipasslem11 30926	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈))) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)))))
21	6, 11, 20	dedth2h 4527	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ ℂ → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))))
22	21	com12 32	. 2 ⊢ (𝐴 ∈ ℂ → ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))))
23	22	3impib 1117	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ifcif 4467 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 · cmul 11034 +_𝑣 cpv 30671 BaseSetcba 30672 ·_𝑠OLD cns 30673 0_veccn0v 30674 ·_𝑖OLDcdip 30786 CPreHil_OLDccphlo 30898

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 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-rmo 3343 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-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-cn 23202 df-cnp 23203 df-t1 23289 df-haus 23290 df-tx 23537 df-hmeo 23730 df-xms 24295 df-ms 24296 df-tms 24297 df-grpo 30579 df-gid 30580 df-ginv 30581 df-gdiv 30582 df-ablo 30631 df-vc 30645 df-nv 30678 df-va 30681 df-ba 30682 df-sm 30683 df-0v 30684 df-vs 30685 df-nmcv 30686 df-ims 30687 df-dip 30787 df-ph 30899

This theorem is referenced by: dipass 30931 ipblnfi 30941

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