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Theorem ipassi 27696

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 6658	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐴𝑆𝐵) = (𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))))
2	1	oveq1d 6665	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → ((𝐴𝑆𝐵)𝑃𝐶) = ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃𝐶))
3		oveq1 6657	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐵𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶))
4	3	oveq2d 6666	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐴 · (𝐵𝑃𝐶)) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶)))
5	2, 4	eqeq12d 2637	. . . . 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 330	. . . 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 6658	. . . . . 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 6658	. . . . . . 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 6666	. . . . . 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 2637	. . . . 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 330	. . . 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 2622	. . . . . 6 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
18	12, 17, 16	elimph 27675	. . . . 5 ⊢ if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) ∈ 𝑋
19	12, 17, 16	elimph 27675	. . . . 5 ⊢ if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)) ∈ 𝑋
20	12, 13, 14, 15, 16, 18, 19	ipasslem11 27695	. . . 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 4140	. . 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 1262	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ifcif 4086 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 · cmul 9941 +_𝑣 cpv 27440 BaseSetcba 27441 ·_𝑠OLD cns 27442 0_veccn0v 27443 ·_𝑖OLDcdip 27555 CPreHil_OLDccphlo 27667

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-cn 21031 df-cnp 21032 df-t1 21118 df-haus 21119 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-dip 27556 df-ph 27668

This theorem is referenced by: dipass 27700 ipblnfi 27711

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