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Mirrors > Home > ILE Home > Th. List > lmod0vid | GIF version |
Description: Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
0vlid.v | ⊢ 𝑉 = (Base‘𝑊) |
0vlid.a | ⊢ + = (+g‘𝑊) |
0vlid.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
lmod0vid | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 13571 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | 0vlid.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | 0vlid.a | . . 3 ⊢ + = (+g‘𝑊) | |
4 | 0vlid.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
5 | 2, 3, 4 | grpid 12949 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) |
6 | 1, 5 | sylan 283 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ‘cfv 5231 (class class class)co 5891 Basecbs 12480 +gcplusg 12555 0gc0g 12727 Grpcgrp 12911 LModclmod 13564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-iota 5193 df-fun 5233 df-fn 5234 df-fv 5239 df-riota 5847 df-ov 5894 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-5 8999 df-6 9000 df-ndx 12483 df-slot 12484 df-base 12486 df-plusg 12568 df-mulr 12569 df-sca 12571 df-vsca 12572 df-0g 12729 df-mgm 12798 df-sgrp 12831 df-mnd 12844 df-grp 12914 df-lmod 13566 |
This theorem is referenced by: lmod0vs 13598 |
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