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Mirrors > Home > MPE Home > Th. List > lmodsubid | Structured version Visualization version GIF version |
Description: Subtraction of a vector from itself. (hvsubid 28803 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodsubeq0.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodsubeq0.o | ⊢ 0 = (0g‘𝑊) |
lmodsubeq0.m | ⊢ − = (-g‘𝑊) |
Ref | Expression |
---|---|
lmodsubid | ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉) → (𝐴 − 𝐴) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 19641 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodsubeq0.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodsubeq0.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | lmodsubeq0.m | . . 3 ⊢ − = (-g‘𝑊) | |
5 | 2, 3, 4 | grpsubid 18183 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉) → (𝐴 − 𝐴) = 0 ) |
6 | 1, 5 | sylan 582 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉) → (𝐴 − 𝐴) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 0gc0g 16713 Grpcgrp 18103 -gcsg 18105 LModclmod 19634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-lmod 19636 |
This theorem is referenced by: lss0cl 19718 ttgbtwnid 26670 |
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