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Theorem dipass 30873

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
ipass.1	⊢ 𝑋 = (BaseSet‘𝑈)
ipass.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
ipass.7	⊢ 𝑃 = (·_𝑖OLD‘𝑈)

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

**Proof of Theorem dipass**
Step	Hyp	Ref	Expression
1		ipass.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
2		fveq2 6906	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
3	1, 2	eqtrid 2786	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑋 = (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
4	3	eleq2d 2824	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))))
5	3	eleq2d 2824	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝐶 ∈ 𝑋 ↔ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))))
6	4, 5	3anbi23d 1438	. . . 4 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ↔ (𝐴 ∈ ℂ ∧ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∧ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))))
7		ipass.4	. . . . . . . . 9 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
8		fveq2 6906	. . . . . . . . 9 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
9	7, 8	eqtrid 2786	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑆 = ( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
10	9	oveqd 7447	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝐴𝑆𝐵) = (𝐴( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐵))
11	10	oveq1d 7445	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ((𝐴𝑆𝐵)𝑃𝐶) = ((𝐴( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐵)𝑃𝐶))
12		ipass.7	. . . . . . . 8 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
13		fveq2 6906	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (·_𝑖OLD‘𝑈) = (·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
14	12, 13	eqtrid 2786	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑃 = (·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
15	14	oveqd 7447	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ((𝐴( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐵)𝑃𝐶) = ((𝐴( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐵)(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶))
16	11, 15	eqtrd 2774	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ((𝐴𝑆𝐵)𝑃𝐶) = ((𝐴( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐵)(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶))
17	14	oveqd 7447	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝐵𝑃𝐶) = (𝐵(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶))
18	17	oveq2d 7446	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝐴 · (𝐵𝑃𝐶)) = (𝐴 · (𝐵(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶)))
19	16, 18	eqeq12d 2750	. . . 4 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)) ↔ ((𝐴( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐵)(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶) = (𝐴 · (𝐵(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶))))
20	6, 19	imbi12d 344	. . 3 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∧ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))) → ((𝐴( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐵)(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶) = (𝐴 · (𝐵(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶)))))
21		eqid 2734	. . . 4 ⊢ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
22		eqid 2734	. . . 4 ⊢ ( +_𝑣 ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = ( +_𝑣 ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
23		eqid 2734	. . . 4 ⊢ ( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = ( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
24		eqid 2734	. . . 4 ⊢ (·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
25		elimphu 30849	. . . 4 ⊢ if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CPreHil_OLD
26	21, 22, 23, 24, 25	ipassi 30869	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∧ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))) → ((𝐴( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐵)(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶) = (𝐴 · (𝐵(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶)))
27	20, 26	dedth 4588	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))))
28	27	imp 406	1 ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (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 1536 ∈ wcel 2105 ifcif 4530 ⟨cop 4636 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 + caddc 11155 · cmul 11157 abscabs 15269 +_𝑣 cpv 30613 BaseSetcba 30614 ·_𝑠OLD cns 30615 ·_𝑖OLDcdip 30728 CPreHil_OLDccphlo 30840

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-cn 23250 df-cnp 23251 df-t1 23337 df-haus 23338 df-tx 23585 df-hmeo 23778 df-xms 24345 df-ms 24346 df-tms 24347 df-grpo 30521 df-gid 30522 df-ginv 30523 df-gdiv 30524 df-ablo 30573 df-vc 30587 df-nv 30620 df-va 30623 df-ba 30624 df-sm 30625 df-0v 30626 df-vs 30627 df-nmcv 30628 df-ims 30629 df-dip 30729 df-ph 30841

This theorem is referenced by: dipassr 30874 dipsubdir 30876 siilem1 30879 hlipass 30941

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