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Mirrors > Home > MPE Home > Th. List > ipasslem9 | Structured version Visualization version GIF version |
Description: Lemma for ipassi 28029. Conclude from ipasslem8 28025 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 6798 | . . . . . 6 ⊢ (𝑤 = 𝐶 → (𝑤𝑆𝐴) = (𝐶𝑆𝐴)) | |
2 | 1 | oveq1d 6806 | . . . . 5 ⊢ (𝑤 = 𝐶 → ((𝑤𝑆𝐴)𝑃𝐵) = ((𝐶𝑆𝐴)𝑃𝐵)) |
3 | oveq1 6798 | . . . . 5 ⊢ (𝑤 = 𝐶 → (𝑤 · (𝐴𝑃𝐵)) = (𝐶 · (𝐴𝑃𝐵))) | |
4 | 2, 3 | oveq12d 6809 | . . . 4 ⊢ (𝑤 = 𝐶 → (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))) = (((𝐶𝑆𝐴)𝑃𝐵) − (𝐶 · (𝐴𝑃𝐵)))) |
5 | eqid 2771 | . . . 4 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) = (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))) | |
6 | ovex 6821 | . . . 4 ⊢ (((𝐶𝑆𝐴)𝑃𝐵) − (𝐶 · (𝐴𝑃𝐵))) ∈ V | |
7 | 4, 5, 6 | fvmpt 6422 | . . 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 28025 | . . . 4 ⊢ (𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))):ℝ⟶{0} |
16 | fvconst 6572 | . . . 4 ⊢ (((𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵)))):ℝ⟶{0} ∧ 𝐶 ∈ ℝ) → ((𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))‘𝐶) = 0) | |
17 | 15, 16 | mpan 670 | . . 3 ⊢ (𝐶 ∈ ℝ → ((𝑤 ∈ ℝ ↦ (((𝑤𝑆𝐴)𝑃𝐵) − (𝑤 · (𝐴𝑃𝐵))))‘𝐶) = 0) |
18 | 7, 17 | eqtr3d 2807 | . 2 ⊢ (𝐶 ∈ ℝ → (((𝐶𝑆𝐴)𝑃𝐵) − (𝐶 · (𝐴𝑃𝐵))) = 0) |
19 | recn 10226 | . . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
20 | 12 | phnvi 28004 | . . . . . 6 ⊢ 𝑈 ∈ NrmCVec |
21 | 8, 10 | nvscl 27814 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ ℂ ∧ 𝐴 ∈ 𝑋) → (𝐶𝑆𝐴) ∈ 𝑋) |
22 | 20, 13, 21 | mp3an13 1563 | . . . . 5 ⊢ (𝐶 ∈ ℂ → (𝐶𝑆𝐴) ∈ 𝑋) |
23 | 8, 11 | dipcl 27900 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶𝑆𝐴) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐶𝑆𝐴)𝑃𝐵) ∈ ℂ) |
24 | 20, 14, 23 | mp3an13 1563 | . . . . 5 ⊢ ((𝐶𝑆𝐴) ∈ 𝑋 → ((𝐶𝑆𝐴)𝑃𝐵) ∈ ℂ) |
25 | 22, 24 | syl 17 | . . . 4 ⊢ (𝐶 ∈ ℂ → ((𝐶𝑆𝐴)𝑃𝐵) ∈ ℂ) |
26 | 8, 11 | dipcl 27900 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) |
27 | 20, 13, 14, 26 | mp3an 1572 | . . . . 5 ⊢ (𝐴𝑃𝐵) ∈ ℂ |
28 | mulcl 10220 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ (𝐴𝑃𝐵) ∈ ℂ) → (𝐶 · (𝐴𝑃𝐵)) ∈ ℂ) | |
29 | 27, 28 | mpan2 671 | . . . 4 ⊢ (𝐶 ∈ ℂ → (𝐶 · (𝐴𝑃𝐵)) ∈ ℂ) |
30 | 25, 29 | subeq0ad 10602 | . . 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 222 | 1 ⊢ (𝐶 ∈ ℝ → ((𝐶𝑆𝐴)𝑃𝐵) = (𝐶 · (𝐴𝑃𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1631 ∈ wcel 2145 {csn 4316 ↦ cmpt 4863 ⟶wf 6025 ‘cfv 6029 (class class class)co 6791 ℂcc 10134 ℝcr 10135 0cc0 10136 · cmul 10141 − cmin 10466 NrmCVeccnv 27772 +𝑣 cpv 27773 BaseSetcba 27774 ·𝑠OLD cns 27775 ·𝑖OLDcdip 27888 CPreHilOLDccphlo 28000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-inf2 8700 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 ax-addf 10215 ax-mulf 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-of 7042 df-om 7211 df-1st 7313 df-2nd 7314 df-supp 7445 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-2o 7712 df-oadd 7715 df-er 7894 df-map 8009 df-ixp 8061 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-fsupp 8430 df-fi 8471 df-sup 8502 df-inf 8503 df-oi 8569 df-card 8963 df-cda 9190 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-4 11281 df-5 11282 df-6 11283 df-7 11284 df-8 11285 df-9 11286 df-n0 11493 df-z 11578 df-dec 11694 df-uz 11887 df-q 11990 df-rp 12029 df-xneg 12144 df-xadd 12145 df-xmul 12146 df-ioo 12377 df-icc 12380 df-fz 12527 df-fzo 12667 df-seq 13002 df-exp 13061 df-hash 13315 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-clim 14420 df-sum 14618 df-struct 16059 df-ndx 16060 df-slot 16061 df-base 16063 df-sets 16064 df-ress 16065 df-plusg 16155 df-mulr 16156 df-starv 16157 df-sca 16158 df-vsca 16159 df-ip 16160 df-tset 16161 df-ple 16162 df-ds 16165 df-unif 16166 df-hom 16167 df-cco 16168 df-rest 16284 df-topn 16285 df-0g 16303 df-gsum 16304 df-topgen 16305 df-pt 16306 df-prds 16309 df-xrs 16363 df-qtop 16368 df-imas 16369 df-xps 16371 df-mre 16447 df-mrc 16448 df-acs 16450 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19946 df-xmet 19947 df-met 19948 df-bl 19949 df-mopn 19950 df-cnfld 19955 df-top 20912 df-topon 20929 df-topsp 20951 df-bases 20964 df-cld 21037 df-ntr 21038 df-cls 21039 df-cn 21245 df-cnp 21246 df-t1 21332 df-haus 21333 df-tx 21579 df-hmeo 21772 df-xms 22338 df-ms 22339 df-tms 22340 df-grpo 27680 df-gid 27681 df-ginv 27682 df-gdiv 27683 df-ablo 27732 df-vc 27747 df-nv 27780 df-va 27783 df-ba 27784 df-sm 27785 df-0v 27786 df-vs 27787 df-nmcv 27788 df-ims 27789 df-dip 27889 df-ph 28001 |
This theorem is referenced by: ipasslem11 28028 |
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