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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 6670 | . . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) | |
3 | 1, 2 | syl5eq 2868 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → 𝑋 = (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) |
4 | 3 | eleq2d 2898 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)))) |
5 | 3 | eleq2d 2898 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐶 ∈ 𝑋 ↔ 𝐶 ∈ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)))) |
6 | 4, 5 | 3anbi23d 1435 | . . . 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 6670 | . . . . . . . . 9 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) | |
9 | 7, 8 | syl5eq 2868 | . . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → 𝑆 = ( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) |
10 | 9 | oveqd 7173 | . . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐴𝑆𝐵) = (𝐴( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐵)) |
11 | 10 | oveq1d 7171 | . . . . . 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 6670 | . . . . . . . 8 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (·𝑖OLD‘𝑈) = (·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) | |
14 | 12, 13 | syl5eq 2868 | . . . . . . 7 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → 𝑃 = (·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))) |
15 | 14 | oveqd 7173 | . . . . . 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 2856 | . . . . 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 7173 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐵𝑃𝐶) = (𝐵(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶)) |
18 | 17 | oveq2d 7172 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝐴 · (𝐵𝑃𝐶)) = (𝐴 · (𝐵(·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉))𝐶))) |
19 | 16, 18 | eqeq12d 2837 | . . . 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 347 | . . 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 2821 | . . . 4 ⊢ (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = (BaseSet‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
22 | eqid 2821 | . . . 4 ⊢ ( +𝑣 ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = ( +𝑣 ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
23 | eqid 2821 | . . . 4 ⊢ ( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = ( ·𝑠OLD ‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
24 | eqid 2821 | . . . 4 ⊢ (·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) = (·𝑖OLD‘if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)) | |
25 | elimphu 28598 | . . . 4 ⊢ if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) ∈ CPreHilOLD | |
26 | 21, 22, 23, 24, 25 | ipassi 28618 | . . 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 4523 | . 2 ⊢ (𝑈 ∈ CPreHilOLD → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶)))) |
28 | 27 | imp 409 | 1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑆𝐵)𝑃𝐶) = (𝐴 · (𝐵𝑃𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ifcif 4467 〈cop 4573 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 + caddc 10540 · cmul 10542 abscabs 14593 +𝑣 cpv 28362 BaseSetcba 28363 ·𝑠OLD cns 28364 ·𝑖OLDcdip 28477 CPreHilOLDccphlo 28589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-cn 21835 df-cnp 21836 df-t1 21922 df-haus 21923 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 df-grpo 28270 df-gid 28271 df-ginv 28272 df-gdiv 28273 df-ablo 28322 df-vc 28336 df-nv 28369 df-va 28372 df-ba 28373 df-sm 28374 df-0v 28375 df-vs 28376 df-nmcv 28377 df-ims 28378 df-dip 28478 df-ph 28590 |
This theorem is referenced by: dipassr 28623 dipsubdir 28625 siilem1 28628 hlipass 28690 |
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