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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 2820 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
2 | eqid 2820 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
3 | 1, 2 | nzrnz 20029 | . . . 4 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
4 | 3 | 3ad2ant1 1128 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (1r‘𝑅) ≠ (0g‘𝑅)) |
5 | uvcn0.u | . . . 4 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
6 | simp1 1131 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝑅 ∈ NzRing) | |
7 | simp2 1132 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝐼 ∈ 𝑊) | |
8 | simp3 1133 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝐽 ∈ 𝐼) | |
9 | 5, 6, 7, 8, 1 | uvcvv1 20929 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐽) = (1r‘𝑅)) |
10 | nzrring 20030 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
11 | 10 | 3ad2ant1 1128 | . . . . . . 7 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝑅 ∈ Ring) |
12 | uvcn0.y | . . . . . . . 8 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
13 | 12, 2 | frlm0 20894 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × {(0g‘𝑅)}) = (0g‘𝑌)) |
14 | 11, 7, 13 | syl2anc 586 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝐼 × {(0g‘𝑅)}) = (0g‘𝑌)) |
15 | uvcn0.0 | . . . . . 6 ⊢ 0 = (0g‘𝑌) | |
16 | 14, 15 | syl6reqr 2874 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 0 = (𝐼 × {(0g‘𝑅)})) |
17 | 16 | fveq1d 6669 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ( 0 ‘𝐽) = ((𝐼 × {(0g‘𝑅)})‘𝐽)) |
18 | fvex 6680 | . . . . . 6 ⊢ (0g‘𝑅) ∈ V | |
19 | 18 | fvconst2 6963 | . . . . 5 ⊢ (𝐽 ∈ 𝐼 → ((𝐼 × {(0g‘𝑅)})‘𝐽) = (0g‘𝑅)) |
20 | 8, 19 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝐼 × {(0g‘𝑅)})‘𝐽) = (0g‘𝑅)) |
21 | 17, 20 | eqtrd 2855 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ( 0 ‘𝐽) = (0g‘𝑅)) |
22 | 4, 9, 21 | 3netr4d 3092 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐽) ≠ ( 0 ‘𝐽)) |
23 | fveq1 6666 | . . 3 ⊢ ((𝑈‘𝐽) = 0 → ((𝑈‘𝐽)‘𝐽) = ( 0 ‘𝐽)) | |
24 | 23 | adantl 484 | . 2 ⊢ (((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ (𝑈‘𝐽) = 0 ) → ((𝑈‘𝐽)‘𝐽) = ( 0 ‘𝐽)) |
25 | 22, 24 | mteqand 3121 | 1 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 {csn 4564 × cxp 5550 ‘cfv 6352 (class class class)co 7153 Basecbs 16479 0gc0g 16709 1rcur 19247 Ringcrg 19293 NzRingcnzr 20026 freeLMod cfrlm 20886 unitVec cuvc 20922 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-oadd 8103 df-er 8286 df-map 8405 df-ixp 8459 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-sup 8903 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-z 11980 df-dec 12097 df-uz 12242 df-fz 12891 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-mulr 16575 df-sca 16577 df-vsca 16578 df-ip 16579 df-tset 16580 df-ple 16581 df-ds 16583 df-hom 16585 df-cco 16586 df-0g 16711 df-prds 16717 df-pws 16719 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-grp 18102 df-minusg 18103 df-sbg 18104 df-subg 18272 df-mgp 19236 df-ur 19248 df-ring 19295 df-subrg 19529 df-lmod 19632 df-lss 19700 df-sra 19940 df-rgmod 19941 df-nzr 20027 df-dsmm 20872 df-frlm 20887 df-uvc 20923 |
This theorem is referenced by: 0prjspnlem 39343 |
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