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Mirrors > Home > MPE Home > Th. List > ipasslem9 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 28214. Conclude from ipasslem8 28210 the inner product associative law for real numbers. (Contributed by NM, 24-Aug-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ip1i.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
ip1i.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
ip1i.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
ip1i.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
ip1i.9 | ⊢ 𝑈 ∈ CPreHilOLD |
ipasslem9.a | ⊢ 𝐴 ∈ 𝑋 |
ipasslem9.b | ⊢ 𝐵 ∈ 𝑋 |
Ref | Expression |
---|---|
ipasslem9 | ⊢ (𝐶 ∈ ℝ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6884 | . . . . . 6 ⊢ (𝑤 = 𝐶 → (𝑤𝑆𝐴) = (𝐶𝑆𝐴)) | |
2 | 1 | oveq1d 6892 | . . . . 5 ⊢ (𝑤 = 𝐶 → ((𝑤𝑆𝐴)𝑃𝐵) = ((𝐶𝑆𝐴)𝑃𝐵)) |
3 | oveq1 6884 | . . . . 5 ⊢ (𝑤 = 𝐶 → (𝑤 · (𝐴𝑃𝐵)) = (𝐶 · (𝐴𝑃𝐵))) | |
4 | 2, 3 | oveq12d 6895 | . . . 4 ⊢ (𝑤 = 𝐶 → (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) = (((𝐶𝑆𝐴)𝑃𝐵) − (𝐶 · (𝐴𝑃𝐵)))) |
5 | eqid 2798 | . . . 4 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) | |
6 | ovex 6909 | . . . 4 ⊢ (((𝐶𝑆𝐴)𝑃𝐵) − (𝐶 · (𝐴𝑃𝐵))) ∈ V | |
7 | 4, 5, 6 | fvmpt 6506 | . . 3 ⊢ (𝐶 ∈ ℝ → ((𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))‘𝐶) = (((𝐶𝑆𝐴)𝑃𝐵) − (𝐶 · (𝐴𝑃𝐵)))) |
8 | ip1i.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
9 | ip1i.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
10 | ip1i.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
11 | ip1i.7 | . . . . 5 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
12 | ip1i.9 | . . . . 5 ⊢ 𝑈 ∈ CPreHilOLD | |
13 | ipasslem9.a | . . . . 5 ⊢ 𝐴 ∈ 𝑋 | |
14 | ipasslem9.b | . . . . 5 ⊢ 𝐵 ∈ 𝑋 | |
15 | 8, 9, 10, 11, 12, 13, 14, 5 | ipasslem8 28210 | . . . 4 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))):ℝ⟶{0} |
16 | fvconst 6658 | . . . 4 ⊢ (((𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))):ℝ⟶{0} ∧ 𝐶 ∈ ℝ) → ((𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))‘𝐶) = 0) | |
17 | 15, 16 | mpan 682 | . . 3 ⊢ (𝐶 ∈ ℝ → ((𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))‘𝐶) = 0) |
18 | 7, 17 | eqtr3d 2834 | . 2 ⊢ (𝐶 ∈ ℝ → (((𝐶𝑆𝐴)𝑃𝐵) − (𝐶 · (𝐴𝑃𝐵))) = 0) |
19 | recn 10313 | . . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
20 | 12 | phnvi 28189 | . . . . . 6 ⊢ 𝑈 ∈ NrmCVec |
21 | 8, 10 | nvscl 27999 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝐶𝑆𝐴) ∈ 𝑋) |
22 | 20, 13, 21 | mp3an13 1577 | . . . . 5 ⊢ (𝐶 ∈ ℂ → (𝐶𝑆𝐴) ∈ 𝑋) |
23 | 8, 11 | dipcl 28085 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐶𝑆𝐴)𝑃𝐵) ∈ ℂ) |
24 | 20, 14, 23 | mp3an13 1577 | . . . . 5 ⊢ ((𝐶𝑆𝐴) ∈ 𝑋 → ((𝐶𝑆𝐴)𝑃𝐵) ∈ ℂ) |
25 | 22, 24 | syl 17 | . . . 4 ⊢ (𝐶 ∈ ℂ → ((𝐶𝑆𝐴)𝑃𝐵) ∈ ℂ) |
26 | 8, 11 | dipcl 28085 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) |
27 | 20, 13, 14, 26 | mp3an 1586 | . . . . 5 ⊢ (𝐴𝑃𝐵) ∈ ℂ |
28 | mulcl 10307 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ (𝐴𝑃𝐵) ∈ ℂ) → (𝐶 · (𝐴𝑃𝐵)) ∈ ℂ) | |
29 | 27, 28 | mpan2 683 | . . . 4 ⊢ (𝐶 ∈ ℂ → (𝐶 · (𝐴𝑃𝐵)) ∈ ℂ) |
30 | 25, 29 | subeq0ad 10693 | . . 3 ⊢ (𝐶 ∈ ℂ → ((((𝐶𝑆𝐴)𝑃𝐵) − (𝐶 · (𝐴𝑃𝐵))) = 0 ↔ ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))) |
31 | 19, 30 | syl 17 | . 2 ⊢ (𝐶 ∈ ℝ → ((((𝐶𝑆𝐴)𝑃𝐵) − (𝐶 · (𝐴𝑃𝐵))) = 0 ↔ ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵)))) |
32 | 18, 31 | mpbid 224 | 1 ⊢ (𝐶 ∈ ℝ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 {csn 4367 ↦ cmpt 4921 ⟶wf 6096 ‘cfv 6100 (class class class)co 6877 ℂcc 10221 ℝcr 10222 0cc0 10223 · cmul 10228 − cmin 10555 NrmCVeccnv 27957 +𝑣 cpv 27958 BaseSetcba 27959 ·𝑠OLD cns 27960 ·𝑖OLDcdip 28073 CPreHilOLDccphlo 28185 |
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 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-inf2 8787 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 ax-pre-sup 10301 ax-addf 10302 ax-mulf 10303 |
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 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-iin 4712 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-se 5271 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-isom 6109 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-of 7130 df-om 7299 df-1st 7400 df-2nd 7401 df-supp 7532 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-2o 7799 df-oadd 7802 df-er 7981 df-map 8096 df-ixp 8148 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-fsupp 8517 df-fi 8558 df-sup 8589 df-inf 8590 df-oi 8656 df-card 9050 df-cda 9277 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-div 10976 df-nn 11312 df-2 11373 df-3 11374 df-4 11375 df-5 11376 df-6 11377 df-7 11378 df-8 11379 df-9 11380 df-n0 11578 df-z 11664 df-dec 11781 df-uz 11928 df-q 12031 df-rp 12072 df-xneg 12190 df-xadd 12191 df-xmul 12192 df-ioo 12425 df-icc 12428 df-fz 12578 df-fzo 12718 df-seq 13053 df-exp 13112 df-hash 13368 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-clim 14557 df-sum 14755 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-hom 16288 df-cco 16289 df-rest 16395 df-topn 16396 df-0g 16414 df-gsum 16415 df-topgen 16416 df-pt 16417 df-prds 16420 df-xrs 16474 df-qtop 16479 df-imas 16480 df-xps 16482 df-mre 16558 df-mrc 16559 df-acs 16561 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-submnd 17648 df-mulg 17854 df-cntz 18059 df-cmn 18507 df-psmet 20057 df-xmet 20058 df-met 20059 df-bl 20060 df-mopn 20061 df-cnfld 20066 df-top 21024 df-topon 21041 df-topsp 21063 df-bases 21076 df-cld 21149 df-ntr 21150 df-cls 21151 df-cn 21357 df-cnp 21358 df-t1 21444 df-haus 21445 df-tx 21691 df-hmeo 21884 df-xms 22450 df-ms 22451 df-tms 22452 df-grpo 27866 df-gid 27867 df-ginv 27868 df-gdiv 27869 df-ablo 27918 df-vc 27932 df-nv 27965 df-va 27968 df-ba 27969 df-sm 27970 df-0v 27971 df-vs 27972 df-nmcv 27973 df-ims 27974 df-dip 28074 df-ph 28186 |
This theorem is referenced by: ipasslem11 28213 |
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