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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdass | Structured version Visualization version GIF version |
Description: Semiring left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdvacl.v | ⊢ 𝑉 = (Base‘𝑊) |
slmdvacl.a | ⊢ + = (+g‘𝑊) |
Ref | Expression |
---|---|
slmdass | ⊢ ((𝑊 ∈ SLMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdmnd 29759 | . 2 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | |
2 | slmdvacl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | slmdvacl.a | . . 3 ⊢ + = (+g‘𝑊) | |
4 | 2, 3 | mndass 17302 | . 2 ⊢ ((𝑊 ∈ Mnd ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
5 | 1, 4 | sylan 488 | 1 ⊢ ((𝑊 ∈ SLMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Mndcmnd 17294 SLModcslmd 29753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-sgrp 17284 df-mnd 17295 df-cmn 18195 df-slmd 29754 |
This theorem is referenced by: (None) |
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