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Mirrors > Home > MPE Home > Th. List > dipass | Structured version Visualization version GIF version |
Description: Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ipass.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
ipass.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
ipass.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
Ref | Expression |
---|---|
dipass | ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipass.1 | . . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | fveq2 6411 | . . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) | |
3 | 1, 2 | syl5eq 2845 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → 𝑋 = (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) |
4 | 3 | eleq2d 2864 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)))) |
5 | 3 | eleq2d 2864 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐶 ∈ 𝑋 ↔ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)))) |
6 | 4, 5 | 3anbi23d 1564 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ↔ (𝐴 ∈ ℂ ∧ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) ∧ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))))) |
7 | ipass.4 | . . . . . . . . 9 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
8 | fveq2 6411 | . . . . . . . . 9 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) | |
9 | 7, 8 | syl5eq 2845 | . . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → 𝑆 = ( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) |
10 | 9 | oveqd 6895 | . . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐴𝑆𝐵) = (𝐴( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐵)) |
11 | 10 | oveq1d 6893 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → ((𝐴𝑆𝐵)𝑃𝐶) = ((𝐴( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐵)𝑃𝐶)) |
12 | ipass.7 | . . . . . . . 8 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
13 | fveq2 6411 | . . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (·𝑖OLD‘𝑈) = (·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) | |
14 | 12, 13 | syl5eq 2845 | . . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → 𝑃 = (·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) |
15 | 14 | oveqd 6895 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → ((𝐴( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐵)𝑃𝐶) = ((𝐴( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐵)(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶)) |
16 | 11, 15 | eqtrd 2833 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → ((𝐴𝑆𝐵)𝑃𝐶) = ((𝐴( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐵)(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶)) |
17 | 14 | oveqd 6895 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐵𝑃𝐶) = (𝐵(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶)) |
18 | 17 | oveq2d 6894 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐴 · (𝐵𝑃𝐶)) = (𝐴 · (𝐵(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶))) |
19 | 16, 18 | eqeq12d 2814 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)) ↔ ((𝐴( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐵)(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶) = (𝐴 · (𝐵(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶)))) |
20 | 6, 19 | imbi12d 336 | . . 3 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) ↔ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) ∧ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) → ((𝐴( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐵)(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶) = (𝐴 · (𝐵(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶))))) |
21 | eqid 2799 | . . . 4 ⊢ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
22 | eqid 2799 | . . . 4 ⊢ ( +𝑣 ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = ( +𝑣 ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
23 | eqid 2799 | . . . 4 ⊢ ( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = ( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
24 | eqid 2799 | . . . 4 ⊢ (·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = (·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
25 | elimphu 28201 | . . . 4 ⊢ if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) ∈ CPreHilOLD | |
26 | 21, 22, 23, 24, 25 | ipassi 28221 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) ∧ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) → ((𝐴( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐵)(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶) = (𝐴 · (𝐵(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶))) |
27 | 20, 26 | dedth 4333 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))) |
28 | 27 | imp 396 | 1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ifcif 4277 〈cop 4374 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 + caddc 10227 · cmul 10229 abscabs 14315 +𝑣 cpv 27965 BaseSetcba 27966 ·𝑠OLD cns 27967 ·𝑖OLDcdip 28080 CPreHilOLDccphlo 28192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-icc 12431 df-fz 12581 df-fzo 12721 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-clim 14560 df-sum 14758 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-hom 16291 df-cco 16292 df-rest 16398 df-topn 16399 df-0g 16417 df-gsum 16418 df-topgen 16419 df-pt 16420 df-prds 16423 df-xrs 16477 df-qtop 16482 df-imas 16483 df-xps 16485 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-mulg 17857 df-cntz 18062 df-cmn 18510 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-cnfld 20069 df-top 21027 df-topon 21044 df-topsp 21066 df-bases 21079 df-cld 21152 df-ntr 21153 df-cls 21154 df-cn 21360 df-cnp 21361 df-t1 21447 df-haus 21448 df-tx 21694 df-hmeo 21887 df-xms 22453 df-ms 22454 df-tms 22455 df-grpo 27873 df-gid 27874 df-ginv 27875 df-gdiv 27876 df-ablo 27925 df-vc 27939 df-nv 27972 df-va 27975 df-ba 27976 df-sm 27977 df-0v 27978 df-vs 27979 df-nmcv 27980 df-ims 27981 df-dip 28081 df-ph 28193 |
This theorem is referenced by: dipassr 28226 dipsubdir 28228 siilem1 28231 hlipass 28294 |
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