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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 24716 | . . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp) |
3 | 2 | 3ad2ant1 1132 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ NrmGrp) |
4 | simp2 1136 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝐾) | |
5 | nlmmul0or.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
6 | eqid 2734 | . . . . . 6 ⊢ (norm‘𝐹) = (norm‘𝐹) | |
7 | 5, 6 | nmcl 24644 | . . . . 5 ⊢ ((𝐹 ∈ NrmGrp ∧ 𝐴 ∈ 𝐾) → ((norm‘𝐹)‘𝐴) ∈ ℝ) |
8 | 3, 4, 7 | syl2anc 584 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝐹)‘𝐴) ∈ ℝ) |
9 | 8 | recnd 11286 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝐹)‘𝐴) ∈ ℂ) |
10 | nlmngp 24713 | . . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
11 | 10 | 3ad2ant1 1132 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ NrmGrp) |
12 | simp3 1137 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
13 | nlmmul0or.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
14 | eqid 2734 | . . . . . 6 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
15 | 13, 14 | nmcl 24644 | . . . . 5 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘𝐵) ∈ ℝ) |
16 | 11, 12, 15 | syl2anc 584 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘𝐵) ∈ ℝ) |
17 | 16 | recnd 11286 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘𝐵) ∈ ℂ) |
18 | 9, 17 | mul0ord 11910 | . 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 24712 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘(𝐴 · 𝐵)) = (((norm‘𝐹)‘𝐴) · ((norm‘𝑊)‘𝐵))) |
21 | 20 | eqeq1d 2736 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘(𝐴 · 𝐵)) = 0 ↔ (((norm‘𝐹)‘𝐴) · ((norm‘𝑊)‘𝐵)) = 0)) |
22 | nlmlmod 24714 | . . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
23 | 13, 1, 19, 5 | lmodvscl 20892 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐴 · 𝐵) ∈ 𝑉) |
24 | 22, 23 | syl3an1 1162 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐴 · 𝐵) ∈ 𝑉) |
25 | nlmmul0or.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
26 | 13, 14, 25 | nmeq0 24646 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 · 𝐵) ∈ 𝑉) → (((norm‘𝑊)‘(𝐴 · 𝐵)) = 0 ↔ (𝐴 · 𝐵) = 0 )) |
27 | 11, 24, 26 | syl2anc 584 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘(𝐴 · 𝐵)) = 0 ↔ (𝐴 · 𝐵) = 0 )) |
28 | 21, 27 | bitr3d 281 | . 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 24646 | . . . 4 ⊢ ((𝐹 ∈ NrmGrp ∧ 𝐴 ∈ 𝐾) → (((norm‘𝐹)‘𝐴) = 0 ↔ 𝐴 = 𝑂)) |
31 | 3, 4, 30 | syl2anc 584 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝐹)‘𝐴) = 0 ↔ 𝐴 = 𝑂)) |
32 | 13, 14, 25 | nmeq0 24646 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘𝐵) = 0 ↔ 𝐵 = 0 )) |
33 | 11, 12, 32 | syl2anc 584 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘𝐵) = 0 ↔ 𝐵 = 0 )) |
34 | 31, 33 | orbi12d 918 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((((norm‘𝐹)‘𝐴) = 0 ∨ ((norm‘𝑊)‘𝐵) = 0) ↔ (𝐴 = 𝑂 ∨ 𝐵 = 0 ))) |
35 | 18, 28, 34 | 3bitr3d 309 | 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 206 ∨ wo 847 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 · cmul 11157 Basecbs 17244 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17485 LModclmod 20874 normcnm 24604 NrmGrpcngp 24605 NrmModcnlm 24608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-0g 17487 df-topgen 17489 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-lmod 20876 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-xms 24345 df-ms 24346 df-nm 24610 df-ngp 24611 df-nrg 24613 df-nlm 24614 |
This theorem is referenced by: (None) |
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