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Theorem ipassi 30932

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 7367	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐴𝑆𝐵) = (𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))))
2	1	oveq1d 7374	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → ((𝐴𝑆𝐵)𝑃𝐶) = ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃𝐶))
3		oveq1 7366	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐵𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶))
4	3	oveq2d 7375	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐴 · (𝐵𝑃𝐶)) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶)))
5	2, 4	eqeq12d 2757	. . . . 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 342	. . . 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 7367	. . . . . 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 7367	. . . . . . 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 7375	. . . . . 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 2757	. . . . 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 342	. . . 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 2741	. . . . . 6 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
18	12, 17, 16	elimph 30911	. . . . 5 ⊢ if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) ∈ 𝑋
19	12, 17, 16	elimph 30911	. . . . 5 ⊢ if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)) ∈ 𝑋
20	12, 13, 14, 15, 16, 18, 19	ipasslem11 30931	. . . 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 4516	. . 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 1123	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ifcif 4456 ‘cfv 6488 (class class class)co 7359 ℂcc 11032 · cmul 11039 +_𝑣 cpv 30676 BaseSetcba 30677 ·_𝑠OLD cns 30678 0_veccn0v 30679 ·_𝑖OLDcdip 30791 CPreHil_OLDccphlo 30903

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-inf2 9557 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112 ax-addf 11113 ax-mulf 11114

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-om 7810 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-icc 13300 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-sum 15644 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19286 df-cmn 19751 df-psmet 21342 df-xmet 21343 df-met 21344 df-bl 21345 df-mopn 21346 df-cnfld 21351 df-top 22880 df-topon 22897 df-topsp 22919 df-bases 22932 df-cld 23005 df-ntr 23006 df-cls 23007 df-cn 23213 df-cnp 23214 df-t1 23300 df-haus 23301 df-tx 23548 df-hmeo 23741 df-xms 24306 df-ms 24307 df-tms 24308 df-grpo 30584 df-gid 30585 df-ginv 30586 df-gdiv 30587 df-ablo 30636 df-vc 30650 df-nv 30683 df-va 30686 df-ba 30687 df-sm 30688 df-0v 30689 df-vs 30690 df-nmcv 30691 df-ims 30692 df-dip 30792 df-ph 30904

This theorem is referenced by: dipass 30936 ipblnfi 30946

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