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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 24517 | . . . . . 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 2731 | . . . . . 6 ⊢ (norm‘𝐹) = (norm‘𝐹) | |
7 | 5, 6 | nmcl 24445 | . . . . 5 ⊢ ((𝐹 ∈ NrmGrp ∧ 𝐴 ∈ 𝐾) → ((norm‘𝐹)‘𝐴) ∈ ℝ) |
8 | 3, 4, 7 | syl2anc 583 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝐹)‘𝐴) ∈ ℝ) |
9 | 8 | recnd 11249 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝐹)‘𝐴) ∈ ℂ) |
10 | nlmngp 24514 | . . . . . 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 2731 | . . . . . 6 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
15 | 13, 14 | nmcl 24445 | . . . . 5 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘𝐵) ∈ ℝ) |
16 | 11, 12, 15 | syl2anc 583 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘𝐵) ∈ ℝ) |
17 | 16 | recnd 11249 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘𝐵) ∈ ℂ) |
18 | 9, 17 | mul0ord 11871 | . 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 24513 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑊)‘(𝐴 · 𝐵)) = (((norm‘𝐹)‘𝐴) · ((norm‘𝑊)‘𝐵))) |
21 | 20 | eqeq1d 2733 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘(𝐴 · 𝐵)) = 0 ↔ (((norm‘𝐹)‘𝐴) · ((norm‘𝑊)‘𝐵)) = 0)) |
22 | nlmlmod 24515 | . . . . 5 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
23 | 13, 1, 19, 5 | lmodvscl 20720 | . . . . 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 24447 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝐴 · 𝐵) ∈ 𝑉) → (((norm‘𝑊)‘(𝐴 · 𝐵)) = 0 ↔ (𝐴 · 𝐵) = 0 )) |
27 | 11, 24, 26 | syl2anc 583 | . . 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 24447 | . . . 4 ⊢ ((𝐹 ∈ NrmGrp ∧ 𝐴 ∈ 𝐾) → (((norm‘𝐹)‘𝐴) = 0 ↔ 𝐴 = 𝑂)) |
31 | 3, 4, 30 | syl2anc 583 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝐹)‘𝐴) = 0 ↔ 𝐴 = 𝑂)) |
32 | 13, 14, 25 | nmeq0 24447 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘𝐵) = 0 ↔ 𝐵 = 0 )) |
33 | 11, 12, 32 | syl2anc 583 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (((norm‘𝑊)‘𝐵) = 0 ↔ 𝐵 = 0 )) |
34 | 31, 33 | orbi12d 916 | . 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 205 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 0cc0 11116 · cmul 11121 Basecbs 17151 Scalarcsca 17207 ·𝑠 cvsca 17208 0gc0g 17392 LModclmod 20702 normcnm 24405 NrmGrpcngp 24406 NrmModcnlm 24409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-0g 17394 df-topgen 17396 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-lmod 20704 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-top 22716 df-topon 22733 df-topsp 22755 df-bases 22769 df-xms 24146 df-ms 24147 df-nm 24411 df-ngp 24412 df-nrg 24414 df-nlm 24415 |
This theorem is referenced by: (None) |
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