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Theorem dipass 31312

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 6882	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
3	1, 2	eqtrid 2809	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑋 = (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
4	3	eleq2d 2848	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))))
5	3	eleq2d 2848	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝐶 ∈ 𝑋 ↔ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))))
6	4, 5	3anbi23d 1467	. . . 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 6882	. . . . . . . . 9 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
9	7, 8	eqtrid 2809	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑆 = ( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
10	9	oveqd 7433	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝐴𝑆𝐵) = (𝐴( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐵))
11	10	oveq1d 7431	. . . . . 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 6882	. . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (·_𝑖OLD‘𝑈) = (·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
14	12, 13	eqtrid 2809	. . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑃 = (·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))
15	14	oveqd 7433	. . . . . 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 2797	. . . . 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 7433	. . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝐵𝑃𝐶) = (𝐵(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶))
18	17	oveq2d 7432	. . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (𝐴 · (𝐵𝑃𝐶)) = (𝐴 · (𝐵(·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))𝐶)))
19	16, 18	eqeq12d 2778	. . . 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 347	. . 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 2762	. . . 4 ⊢ (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (BaseSet‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
22		eqid 2762	. . . 4 ⊢ ( +_𝑣 ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = ( +_𝑣 ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
23		eqid 2762	. . . 4 ⊢ ( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = ( ·_𝑠OLD ‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
24		eqid 2762	. . . 4 ⊢ (·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (·_𝑖OLD‘if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩))
25		elimphu 31288	. . . 4 ⊢ if(𝑈 ∈ CPreHil_OLD, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ CPreHil_OLD
26	21, 22, 23, 24, 25	ipassi 31308	. . 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 4544	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))))
28	27	imp 412	1 ⊢ ((𝑈 ∈ CPreHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ifcif 4485 ⟨cop 4593 ‘cfv 6537 (class class class)co 7416 ℂcc 11125 + caddc 11130 · cmul 11132 abscabs 15323 +_𝑣 cpv 31052 BaseSetcba 31053 ·_𝑠OLD cns 31054 ·_𝑖OLDcdip 31167 CPreHil_OLDccphlo 31279

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13045 df-xneg 13165 df-xadd 13166 df-xmul 13167 df-ioo 13404 df-icc 13407 df-fz 13564 df-fzo 13712 df-seq 14068 df-exp 14128 df-hash 14397 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-clim 15577 df-sum 15776 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-starv 17361 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-unif 17369 df-hom 17370 df-cco 17371 df-rest 17511 df-topn 17512 df-0g 17530 df-gsum 17531 df-topgen 17532 df-pt 17533 df-prds 17536 df-xrs 17592 df-qtop 17597 df-imas 17598 df-xps 17600 df-mre 17674 df-mrc 17675 df-acs 17677 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-submnd 18893 df-mulg 19192 df-cntz 19445 df-cmn 19910 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-cn 23453 df-cnp 23454 df-t1 23540 df-haus 23541 df-tx 23789 df-hmeo 23982 df-xms 24547 df-ms 24548 df-tms 24549 df-grpo 30960 df-gid 30961 df-ginv 30962 df-gdiv 30963 df-ablo 31012 df-vc 31026 df-nv 31059 df-va 31062 df-ba 31063 df-sm 31064 df-0v 31065 df-vs 31066 df-nmcv 31067 df-ims 31068 df-dip 31168 df-ph 31280

This theorem is used by: dipassr 31313 dipsubdir 31315 siilem1 31318 hlipass 31380

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