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Mirrors > Home > ILE Home > Th. List > abl32 | GIF version |
Description: Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
ablcom.b | ⊢ 𝐵 = (Base‘𝐺) |
ablcom.p | ⊢ + = (+g‘𝐺) |
abl32.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
abl32.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
abl32.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
abl32.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
abl32 | ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abl32.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablcmn 12909 | . . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
4 | abl32.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | abl32.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | abl32.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
7 | ablcom.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
8 | ablcom.p | . . 3 ⊢ + = (+g‘𝐺) | |
9 | 7, 8 | cmn32 12921 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
10 | 3, 4, 5, 6, 9 | syl13anc 1240 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 +gcplusg 12505 CMndccmn 12902 Abelcabl 12903 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-ov 5871 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-sgrp 12687 df-mnd 12697 df-cmn 12904 df-abl 12905 |
This theorem is referenced by: (None) |
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