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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmd0vlid | Structured version Visualization version GIF version |
Description: Left identity law for the zero vector. (hvaddlid 30826 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmd0vlid.v | ⊢ 𝑉 = (Base‘𝑊) |
slmd0vlid.a | ⊢ + = (+g‘𝑊) |
slmd0vlid.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
slmd0vlid | ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdmnd 32907 | . 2 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | |
2 | slmd0vlid.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | slmd0vlid.a | . . 3 ⊢ + = (+g‘𝑊) | |
4 | slmd0vlid.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
5 | 2, 3, 4 | mndlid 18707 | . 2 ⊢ ((𝑊 ∈ Mnd ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) |
6 | 1, 5 | sylan 579 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 +gcplusg 17226 0gc0g 17414 Mndcmnd 18687 SLModcslmd 32901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-riota 7370 df-ov 7417 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-cmn 19730 df-slmd 32902 |
This theorem is referenced by: (None) |
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