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Mirrors > Home > MPE Home > Th. List > uvcf1 | Structured version Visualization version GIF version |
Description: In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
uvcff.u | ⊢ 𝑈 = (𝑅 unitVec 𝐼) |
uvcff.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
uvcff.b | ⊢ 𝐵 = (Base‘𝑌) |
Ref | Expression |
---|---|
uvcf1 | ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nzrring 20414 | . . 3 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
2 | uvcff.u | . . . 4 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
3 | uvcff.y | . . . 4 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
4 | uvcff.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
5 | 2, 3, 4 | uvcff 21675 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼⟶𝐵) |
6 | 1, 5 | sylan 579 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼⟶𝐵) |
7 | eqid 2724 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | eqid 2724 | . . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | 7, 8 | nzrnz 20413 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
10 | 9 | ad3antrrr 727 | . . . . . . 7 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → (1r‘𝑅) ≠ (0g‘𝑅)) |
11 | 1 | ad3antrrr 727 | . . . . . . . 8 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → 𝑅 ∈ Ring) |
12 | simpllr 773 | . . . . . . . 8 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → 𝐼 ∈ 𝑊) | |
13 | simplrl 774 | . . . . . . . 8 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → 𝑖 ∈ 𝐼) | |
14 | 2, 11, 12, 13, 7 | uvcvv1 21673 | . . . . . . 7 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → ((𝑈‘𝑖)‘𝑖) = (1r‘𝑅)) |
15 | simplrr 775 | . . . . . . . 8 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → 𝑗 ∈ 𝐼) | |
16 | simpr 484 | . . . . . . . . 9 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → 𝑖 ≠ 𝑗) | |
17 | 16 | necomd 2988 | . . . . . . . 8 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → 𝑗 ≠ 𝑖) |
18 | 2, 11, 12, 15, 13, 17, 8 | uvcvv0 21674 | . . . . . . 7 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → ((𝑈‘𝑗)‘𝑖) = (0g‘𝑅)) |
19 | 10, 14, 18 | 3netr4d 3010 | . . . . . 6 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → ((𝑈‘𝑖)‘𝑖) ≠ ((𝑈‘𝑗)‘𝑖)) |
20 | fveq1 6881 | . . . . . . 7 ⊢ ((𝑈‘𝑖) = (𝑈‘𝑗) → ((𝑈‘𝑖)‘𝑖) = ((𝑈‘𝑗)‘𝑖)) | |
21 | 20 | necon3i 2965 | . . . . . 6 ⊢ (((𝑈‘𝑖)‘𝑖) ≠ ((𝑈‘𝑗)‘𝑖) → (𝑈‘𝑖) ≠ (𝑈‘𝑗)) |
22 | 19, 21 | syl 17 | . . . . 5 ⊢ ((((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) ∧ 𝑖 ≠ 𝑗) → (𝑈‘𝑖) ≠ (𝑈‘𝑗)) |
23 | 22 | ex 412 | . . . 4 ⊢ (((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) → (𝑖 ≠ 𝑗 → (𝑈‘𝑖) ≠ (𝑈‘𝑗))) |
24 | 23 | necon4d 2956 | . . 3 ⊢ (((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼)) → ((𝑈‘𝑖) = (𝑈‘𝑗) → 𝑖 = 𝑗)) |
25 | 24 | ralrimivva 3192 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑈‘𝑖) = (𝑈‘𝑗) → 𝑖 = 𝑗)) |
26 | dff13 7247 | . 2 ⊢ (𝑈:𝐼–1-1→𝐵 ↔ (𝑈:𝐼⟶𝐵 ∧ ∀𝑖 ∈ 𝐼 ∀𝑗 ∈ 𝐼 ((𝑈‘𝑖) = (𝑈‘𝑗) → 𝑖 = 𝑗))) | |
27 | 6, 25, 26 | sylanbrc 582 | 1 ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ⟶wf 6530 –1-1→wf1 6531 ‘cfv 6534 (class class class)co 7402 Basecbs 17149 0gc0g 17390 1rcur 20082 Ringcrg 20134 NzRingcnzr 20410 freeLMod cfrlm 21630 unitVec cuvc 21666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-hom 17226 df-cco 17227 df-0g 17392 df-prds 17398 df-pws 17400 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-mgp 20036 df-ur 20083 df-ring 20136 df-nzr 20411 df-sra 21017 df-rgmod 21018 df-dsmm 21616 df-frlm 21631 df-uvc 21667 |
This theorem is referenced by: frlmlbs 21681 uvcf1o 21730 frlmdim 33204 |
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