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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 2736 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2736 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | nzrnz 20576 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
4 | 3 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (1r‘𝑅) ≠ (0g‘𝑅)) |
5 | uvcn0.u | . . . . 5 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
6 | simp1 1136 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝑅 ∈ NzRing) | |
7 | simp2 1137 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝐼 ∈ 𝑊) | |
8 | simp3 1138 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝐽 ∈ 𝐼) | |
9 | 5, 6, 7, 8, 1 | uvcvv1 21041 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐽) = (1r‘𝑅)) |
10 | uvcn0.y | . . . . 5 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
11 | nzrring 20577 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
12 | 11 | 3ad2ant1 1133 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝑅 ∈ Ring) |
13 | 10, 2, 12, 7, 8 | frlm0vald 40299 | . . . 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 6803 | . . . 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 1087 = wceq 1539 ∈ wcel 2104 ≠ wne 2941 ‘cfv 6458 (class class class)co 7307 Basecbs 16957 0gc0g 17195 1rcur 19782 Ringcrg 19828 NzRingcnzr 20573 freeLMod cfrlm 20998 unitVec cuvc 21034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-sup 9245 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-fz 13286 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-sca 17023 df-vsca 17024 df-ip 17025 df-tset 17026 df-ple 17027 df-ds 17029 df-hom 17031 df-cco 17032 df-0g 17197 df-prds 17203 df-pws 17205 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-grp 18625 df-minusg 18626 df-sbg 18627 df-subg 18797 df-mgp 19766 df-ur 19783 df-ring 19830 df-subrg 20067 df-lmod 20170 df-lss 20239 df-sra 20479 df-rgmod 20480 df-nzr 20574 df-dsmm 20984 df-frlm 20999 df-uvc 21035 |
This theorem is referenced by: 0prjspnlem 40497 |
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