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Mirrors > Home > MPE Home > Th. List > lmodass | Structured version Visualization version GIF version |
Description: Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvacl.v | ⊢ 𝑉 = (Base‘𝑊) |
lmodvacl.a | ⊢ + = (+g‘𝑊) |
Ref | Expression |
---|---|
lmodass | ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 20744 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | lmodvacl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmodvacl.a | . . 3 ⊢ + = (+g‘𝑊) | |
4 | 2, 3 | grpass 18893 | . 2 ⊢ ((𝑊 ∈ Grp ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
5 | 1, 4 | sylan 579 | 1 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 +gcplusg 17227 Grpcgrp 18884 LModclmod 20737 |
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-ext 2699 ax-nul 5301 |
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-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-iota 6495 df-fv 6551 df-ov 7418 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-lmod 20739 |
This theorem is referenced by: lmodvneg1 20782 lmodcom 20785 baerlem5alem1 41176 mapdh6gN 41210 mapdh6hN 41211 hdmap1l6g 41284 hdmap1l6h 41285 |
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