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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 6645 | . . . . . . 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 2875 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)))) |
5 | 3 | eleq2d 2875 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐶 ∈ 𝑋 ↔ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)))) |
6 | 4, 5 | 3anbi23d 1436 | . . . 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 6645 | . . . . . . . . 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 7152 | . . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐴𝑆𝐵) = (𝐴( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐵)) |
11 | 10 | oveq1d 7150 | . . . . . 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 6645 | . . . . . . . 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 7152 | . . . . . 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 7152 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐵𝑃𝐶) = (𝐵(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶)) |
18 | 17 | oveq2d 7151 | . . . . 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 348 | . . 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 2798 | . . . 4 ⊢ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
22 | eqid 2798 | . . . 4 ⊢ ( +𝑣 ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = ( +𝑣 ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
23 | eqid 2798 | . . . 4 ⊢ ( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = ( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
24 | eqid 2798 | . . . 4 ⊢ (·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = (·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
25 | elimphu 28604 | . . . 4 ⊢ if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) ∈ CPreHilOLD | |
26 | 21, 22, 23, 24, 25 | ipassi 28624 | . . 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 4481 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))) |
28 | 27 | imp 410 | 1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ifcif 4425 〈cop 4531 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 + caddc 10529 · cmul 10531 abscabs 14585 +𝑣 cpv 28368 BaseSetcba 28369 ·𝑠OLD cns 28370 ·𝑖OLDcdip 28483 CPreHilOLDccphlo 28595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-cn 21832 df-cnp 21833 df-t1 21919 df-haus 21920 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-dip 28484 df-ph 28596 |
This theorem is referenced by: dipassr 28629 dipsubdir 28631 siilem1 28634 hlipass 28696 |
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