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Theorem ipassi 30916

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 7366	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐴𝑆𝐵) = (𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))))
2	1	oveq1d 7373	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → ((𝐴𝑆𝐵)𝑃𝐶) = ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃𝐶))
3		oveq1 7365	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐵𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶))
4	3	oveq2d 7374	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐴 · (𝐵𝑃𝐶)) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶)))
5	2, 4	eqeq12d 2752	. . . . 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 7366	. . . . . 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 7366	. . . . . . 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 7374	. . . . . 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 2752	. . . . 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 2736	. . . . . 6 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
18	12, 17, 16	elimph 30895	. . . . 5 ⊢ if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) ∈ 𝑋
19	12, 17, 16	elimph 30895	. . . . 5 ⊢ if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)) ∈ 𝑋
20	12, 13, 14, 15, 16, 18, 19	ipasslem11 30915	. . . 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 4539	. . 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 1116	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ifcif 4479 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 · cmul 11031 +_𝑣 cpv 30660 BaseSetcba 30661 ·_𝑠OLD cns 30662 0_veccn0v 30663 ·_𝑖OLDcdip 30775 CPreHil_OLDccphlo 30887

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-icc 13268 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-cn 23171 df-cnp 23172 df-t1 23258 df-haus 23259 df-tx 23506 df-hmeo 23699 df-xms 24264 df-ms 24265 df-tms 24266 df-grpo 30568 df-gid 30569 df-ginv 30570 df-gdiv 30571 df-ablo 30620 df-vc 30634 df-nv 30667 df-va 30670 df-ba 30671 df-sm 30672 df-0v 30673 df-vs 30674 df-nmcv 30675 df-ims 30676 df-dip 30776 df-ph 30888

This theorem is referenced by: dipass 30920 ipblnfi 30930

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