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Mirrors > Home > MPE Home > Th. List > ip2eqi | Structured version Visualization version GIF version |
Description: Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ip2eqi.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
ip2eqi.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
ip2eqi.u | ⊢ 𝑈 ∈ CPreHilOLD |
Ref | Expression |
---|---|
ip2eqi | ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ip2eqi.u | . . . . . 6 ⊢ 𝑈 ∈ CPreHilOLD | |
2 | 1 | phnvi 30845 | . . . . 5 ⊢ 𝑈 ∈ NrmCVec |
3 | ip2eqi.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | eqid 2735 | . . . . . 6 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
5 | 3, 4 | nvmcl 30675 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋) |
6 | 2, 5 | mp3an1 1447 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋) |
7 | oveq1 7438 | . . . . . 6 ⊢ (𝑥 = (𝐴( −𝑣 ‘𝑈)𝐵) → (𝑥𝑃𝐴) = ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴)) | |
8 | oveq1 7438 | . . . . . 6 ⊢ (𝑥 = (𝐴( −𝑣 ‘𝑈)𝐵) → (𝑥𝑃𝐵) = ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵)) | |
9 | 7, 8 | eqeq12d 2751 | . . . . 5 ⊢ (𝑥 = (𝐴( −𝑣 ‘𝑈)𝐵) → ((𝑥𝑃𝐴) = (𝑥𝑃𝐵) ↔ ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) = ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵))) |
10 | 9 | rspcv 3618 | . . . 4 ⊢ ((𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋 → (∀𝑥 ∈ 𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) = ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵))) |
11 | 6, 10 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) = ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵))) |
12 | simpl 482 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
13 | simpr 484 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋) | |
14 | ip2eqi.7 | . . . . . . . . 9 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
15 | 3, 4, 14 | dipsubdi 30878 | . . . . . . . 8 ⊢ ((𝑈 ∈ CPreHilOLD ∧ ((𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃(𝐴( −𝑣 ‘𝑈)𝐵)) = (((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) − ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵))) |
16 | 1, 15 | mpan 690 | . . . . . . 7 ⊢ (((𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃(𝐴( −𝑣 ‘𝑈)𝐵)) = (((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) − ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵))) |
17 | 6, 12, 13, 16 | syl3anc 1370 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃(𝐴( −𝑣 ‘𝑈)𝐵)) = (((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) − ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵))) |
18 | 17 | eqeq1d 2737 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴( −𝑣 ‘𝑈)𝐵)𝑃(𝐴( −𝑣 ‘𝑈)𝐵)) = 0 ↔ (((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) − ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵)) = 0)) |
19 | eqid 2735 | . . . . . . . 8 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
20 | 3, 19, 14 | ipz 30748 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋) → (((𝐴( −𝑣 ‘𝑈)𝐵)𝑃(𝐴( −𝑣 ‘𝑈)𝐵)) = 0 ↔ (𝐴( −𝑣 ‘𝑈)𝐵) = (0vec‘𝑈))) |
21 | 2, 20 | mpan 690 | . . . . . 6 ⊢ ((𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋 → (((𝐴( −𝑣 ‘𝑈)𝐵)𝑃(𝐴( −𝑣 ‘𝑈)𝐵)) = 0 ↔ (𝐴( −𝑣 ‘𝑈)𝐵) = (0vec‘𝑈))) |
22 | 6, 21 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴( −𝑣 ‘𝑈)𝐵)𝑃(𝐴( −𝑣 ‘𝑈)𝐵)) = 0 ↔ (𝐴( −𝑣 ‘𝑈)𝐵) = (0vec‘𝑈))) |
23 | 18, 22 | bitr3d 281 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) − ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵)) = 0 ↔ (𝐴( −𝑣 ‘𝑈)𝐵) = (0vec‘𝑈))) |
24 | 3, 14 | dipcl 30741 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) ∈ ℂ) |
25 | 2, 24 | mp3an1 1447 | . . . . . 6 ⊢ (((𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) ∈ ℂ) |
26 | 6, 12, 25 | syl2anc 584 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) ∈ ℂ) |
27 | 3, 14 | dipcl 30741 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵) ∈ ℂ) |
28 | 2, 27 | mp3an1 1447 | . . . . . 6 ⊢ (((𝐴( −𝑣 ‘𝑈)𝐵) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵) ∈ ℂ) |
29 | 6, 28 | sylancom 588 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵) ∈ ℂ) |
30 | 26, 29 | subeq0ad 11628 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) − ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵)) = 0 ↔ ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) = ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵))) |
31 | 3, 4, 19 | nvmeq0 30687 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴( −𝑣 ‘𝑈)𝐵) = (0vec‘𝑈) ↔ 𝐴 = 𝐵)) |
32 | 2, 31 | mp3an1 1447 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴( −𝑣 ‘𝑈)𝐵) = (0vec‘𝑈) ↔ 𝐴 = 𝐵)) |
33 | 23, 30, 32 | 3bitr3d 309 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐴) = ((𝐴( −𝑣 ‘𝑈)𝐵)𝑃𝐵) ↔ 𝐴 = 𝐵)) |
34 | 11, 33 | sylibd 239 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) → 𝐴 = 𝐵)) |
35 | oveq2 7439 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑥𝑃𝐴) = (𝑥𝑃𝐵)) | |
36 | 35 | ralrimivw 3148 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵)) |
37 | 34, 36 | impbid1 225 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑥𝑃𝐴) = (𝑥𝑃𝐵) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 − cmin 11490 NrmCVeccnv 30613 BaseSetcba 30615 0veccn0v 30617 −𝑣 cnsb 30618 ·𝑖OLDcdip 30729 CPreHilOLDccphlo 30841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-cn 23251 df-cnp 23252 df-t1 23338 df-haus 23339 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 df-grpo 30522 df-gid 30523 df-ginv 30524 df-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629 df-ims 30630 df-dip 30730 df-ph 30842 |
This theorem is referenced by: phoeqi 30886 |
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