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Mirrors > Home > MPE Home > Th. List > nlmmul0or | Structured version Visualization version GIF version |
Description: If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (Revised by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nlmmul0or.v | ⊢ 𝑉 = (Base‘𝑊) |
nlmmul0or.s | ⊢ · = ( ·𝑠 ‘𝑊) |
nlmmul0or.z | ⊢ 0 = (0g‘𝑊) |
nlmmul0or.f | ⊢ 𝐹 = (Scalar‘𝑊) |
nlmmul0or.k | ⊢ 𝐾 = (Base‘𝐹) |
nlmmul0or.o | ⊢ 𝑂 = (0g‘𝐹) |
Ref | Expression |
---|---|
nlmmul0or | ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 𝑂 ∨ 𝐵 = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nlmmul0or.f | . . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | nlmngp2 23283 | . . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp) |
3 | 2 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ NrmGrp) |
4 | simp2 1133 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝐾) | |
5 | nlmmul0or.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
6 | eqid 2821 | . . . . . 6 ⊢ (norm‘𝐹) = (norm‘𝐹) | |
7 | 5, 6 | nmcl 23219 | . . . . 5 ⊢ ((𝐹 ∈ NrmGrp ∧ 𝐴 ∈ 𝐾) → ((norm‘𝐹)‘𝐴) ∈ ℝ) |
8 | 3, 4, 7 | syl2anc 586 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝐹)‘𝐴) ∈ ℝ) |
9 | 8 | recnd 10663 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝐹)‘𝐴) ∈ ℂ) |
10 | nlmngp 23280 | . . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
11 | 10 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ NrmGrp) |
12 | simp3 1134 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
13 | nlmmul0or.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
14 | eqid 2821 | . . . . . 6 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
15 | 13, 14 | nmcl 23219 | . . . . 5 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘𝐵) ∈ ℝ) |
16 | 11, 12, 15 | syl2anc 586 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘𝐵) ∈ ℝ) |
17 | 16 | recnd 10663 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘𝐵) ∈ ℂ) |
18 | 9, 17 | mul0ord 11284 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((((norm‘𝐹)‘𝐴) · ((norm‘𝑊)‘𝐵)) = 0 ↔ (((norm‘𝐹)‘𝐴) = 0 ∨ ((norm‘𝑊)‘𝐵) = 0))) |
19 | nlmmul0or.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
20 | 13, 14, 19, 1, 5, 6 | nmvs 23279 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘(𝐴 · 𝐵)) = (((norm‘𝐹)‘𝐴) · ((norm‘𝑊)‘𝐵))) |
21 | 20 | eqeq1d 2823 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘(𝐴 · 𝐵)) = 0 ↔ (((norm‘𝐹)‘𝐴) · ((norm‘𝑊)‘𝐵)) = 0)) |
22 | nlmlmod 23281 | . . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
23 | 13, 1, 19, 5 | lmodvscl 19645 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐴 · 𝐵) ∈ 𝑉) |
24 | 22, 23 | syl3an1 1159 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐴 · 𝐵) ∈ 𝑉) |
25 | nlmmul0or.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
26 | 13, 14, 25 | nmeq0 23221 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 · 𝐵) ∈ 𝑉) → (((norm‘𝑊)‘(𝐴 · 𝐵)) = 0 ↔ (𝐴 · 𝐵) = 0 )) |
27 | 11, 24, 26 | syl2anc 586 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘(𝐴 · 𝐵)) = 0 ↔ (𝐴 · 𝐵) = 0 )) |
28 | 21, 27 | bitr3d 283 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((((norm‘𝐹)‘𝐴) · ((norm‘𝑊)‘𝐵)) = 0 ↔ (𝐴 · 𝐵) = 0 )) |
29 | nlmmul0or.o | . . . . 5 ⊢ 𝑂 = (0g‘𝐹) | |
30 | 5, 6, 29 | nmeq0 23221 | . . . 4 ⊢ ((𝐹 ∈ NrmGrp ∧ 𝐴 ∈ 𝐾) → (((norm‘𝐹)‘𝐴) = 0 ↔ 𝐴 = 𝑂)) |
31 | 3, 4, 30 | syl2anc 586 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝐹)‘𝐴) = 0 ↔ 𝐴 = 𝑂)) |
32 | 13, 14, 25 | nmeq0 23221 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘𝐵) = 0 ↔ 𝐵 = 0 )) |
33 | 11, 12, 32 | syl2anc 586 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘𝐵) = 0 ↔ 𝐵 = 0 )) |
34 | 31, 33 | orbi12d 915 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((((norm‘𝐹)‘𝐴) = 0 ∨ ((norm‘𝑊)‘𝐵) = 0) ↔ (𝐴 = 𝑂 ∨ 𝐵 = 0 ))) |
35 | 18, 28, 34 | 3bitr3d 311 | 1 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 𝑂 ∨ 𝐵 = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 0cc0 10531 · cmul 10536 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 LModclmod 19628 normcnm 23180 NrmGrpcngp 23181 NrmModcnlm 23184 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-0g 16709 df-topgen 16711 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-lmod 19630 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-xms 22924 df-ms 22925 df-nm 23186 df-ngp 23187 df-nrg 23189 df-nlm 23190 |
This theorem is referenced by: (None) |
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