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Mirrors > Home > MPE Home > Th. List > Mathboxes > uvcn0 | Structured version Visualization version GIF version |
Description: A unit vector is nonzero. (Contributed by Steven Nguyen, 16-Jul-2023.) |
Ref | Expression |
---|---|
uvcn0.u | ⊢ 𝑈 = (𝑅 unitVec 𝐼) |
uvcn0.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
uvcn0.b | ⊢ 𝐵 = (Base‘𝑌) |
uvcn0.0 | ⊢ 0 = (0g‘𝑌) |
Ref | Expression |
---|---|
uvcn0 | ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2733 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | nzrnz 20272 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
4 | 3 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (1r‘𝑅) ≠ (0g‘𝑅)) |
5 | uvcn0.u | . . . . 5 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
6 | simp1 1137 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝑅 ∈ NzRing) | |
7 | simp2 1138 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝐼 ∈ 𝑊) | |
8 | simp3 1139 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝐽 ∈ 𝐼) | |
9 | 5, 6, 7, 8, 1 | uvcvv1 21317 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐽) = (1r‘𝑅)) |
10 | uvcn0.y | . . . . 5 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
11 | nzrring 20273 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
12 | 11 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝑅 ∈ Ring) |
13 | 10, 2, 12, 7, 8 | frlm0vald 41013 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((0g‘𝑌)‘𝐽) = (0g‘𝑅)) |
14 | 4, 9, 13 | 3netr4d 3019 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐽) ≠ ((0g‘𝑌)‘𝐽)) |
15 | fveq1 6880 | . . . 4 ⊢ ((𝑈‘𝐽) = (0g‘𝑌) → ((𝑈‘𝐽)‘𝐽) = ((0g‘𝑌)‘𝐽)) | |
16 | 15 | necon3i 2974 | . . 3 ⊢ (((𝑈‘𝐽)‘𝐽) ≠ ((0g‘𝑌)‘𝐽) → (𝑈‘𝐽) ≠ (0g‘𝑌)) |
17 | 14, 16 | syl 17 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) ≠ (0g‘𝑌)) |
18 | uvcn0.0 | . . 3 ⊢ 0 = (0g‘𝑌) | |
19 | 18 | a1i 11 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 0 = (0g‘𝑌)) |
20 | 17, 19 | neeqtrrd 3016 | 1 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 0gc0g 17372 1rcur 19987 Ringcrg 20038 NzRingcnzr 20269 freeLMod cfrlm 21274 unitVec cuvc 21310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9424 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-fz 13472 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-hom 17208 df-cco 17209 df-0g 17374 df-prds 17380 df-pws 17382 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-grp 18809 df-minusg 18810 df-sbg 18811 df-subg 18988 df-mgp 19971 df-ur 19988 df-ring 20040 df-nzr 20270 df-subrg 20338 df-lmod 20450 df-lss 20520 df-sra 20762 df-rgmod 20763 df-dsmm 21260 df-frlm 21275 df-uvc 21311 |
This theorem is referenced by: 0prjspnlem 41247 |
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