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Theorem ipassi 30776

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 7397	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐴𝑆𝐵) = (𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))))
2	1	oveq1d 7404	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → ((𝐴𝑆𝐵)𝑃𝐶) = ((𝐴𝑆if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)))𝑃𝐶))
3		oveq1 7396	. . . . . . 7 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐵𝑃𝐶) = (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶))
4	3	oveq2d 7405	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) → (𝐴 · (𝐵𝑃𝐶)) = (𝐴 · (if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈))𝑃𝐶)))
5	2, 4	eqeq12d 2746	. . . . 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 7397	. . . . . 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 7397	. . . . . . 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 7405	. . . . . 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 2746	. . . . 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 2730	. . . . . 6 ⊢ (0_vec‘𝑈) = (0_vec‘𝑈)
18	12, 17, 16	elimph 30755	. . . . 5 ⊢ if(𝐵 ∈ 𝑋, 𝐵, (0_vec‘𝑈)) ∈ 𝑋
19	12, 17, 16	elimph 30755	. . . . 5 ⊢ if(𝐶 ∈ 𝑋, 𝐶, (0_vec‘𝑈)) ∈ 𝑋
20	12, 13, 14, 15, 16, 18, 19	ipasslem11 30775	. . . 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 4550	. . 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 1540 ∈ wcel 2109 ifcif 4490 ‘cfv 6513 (class class class)co 7389 ℂcc 11072 · cmul 11079 +_𝑣 cpv 30520 BaseSetcba 30521 ·_𝑠OLD cns 30522 0_veccn0v 30523 ·_𝑖OLDcdip 30635 CPreHil_OLDccphlo 30747

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-inf2 9600 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153 ax-mulf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-er 8673 df-map 8803 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-fi 9368 df-sup 9399 df-inf 9400 df-oi 9469 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-uz 12800 df-q 12914 df-rp 12958 df-xneg 13078 df-xadd 13079 df-xmul 13080 df-ioo 13316 df-icc 13319 df-fz 13475 df-fzo 13622 df-seq 13973 df-exp 14033 df-hash 14302 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-clim 15460 df-sum 15659 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17391 df-topn 17392 df-0g 17410 df-gsum 17411 df-topgen 17412 df-pt 17413 df-prds 17416 df-xrs 17471 df-qtop 17476 df-imas 17477 df-xps 17479 df-mre 17553 df-mrc 17554 df-acs 17556 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18717 df-mulg 19006 df-cntz 19255 df-cmn 19718 df-psmet 21262 df-xmet 21263 df-met 21264 df-bl 21265 df-mopn 21266 df-cnfld 21271 df-top 22787 df-topon 22804 df-topsp 22826 df-bases 22839 df-cld 22912 df-ntr 22913 df-cls 22914 df-cn 23120 df-cnp 23121 df-t1 23207 df-haus 23208 df-tx 23455 df-hmeo 23648 df-xms 24214 df-ms 24215 df-tms 24216 df-grpo 30428 df-gid 30429 df-ginv 30430 df-gdiv 30431 df-ablo 30480 df-vc 30494 df-nv 30527 df-va 30530 df-ba 30531 df-sm 30532 df-0v 30533 df-vs 30534 df-nmcv 30535 df-ims 30536 df-dip 30636 df-ph 30748

This theorem is referenced by: dipass 30780 ipblnfi 30790

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