![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ablsub32 | Structured version Visualization version GIF version |
Description: Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.) |
Ref | Expression |
---|---|
ablnncan.b | ⊢ 𝐵 = (Base‘𝐺) |
ablnncan.m | ⊢ − = (-g‘𝐺) |
ablnncan.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablnncan.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ablnncan.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ablsub32.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
ablsub32 | ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑍) − 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablnncan.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablnncan.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
3 | ablsub32.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
4 | ablnncan.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
5 | eqid 2737 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 4, 5 | ablcom 19539 | . . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)𝑌)) |
7 | 1, 2, 3, 6 | syl3anc 1371 | . . 3 ⊢ (𝜑 → (𝑌(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)𝑌)) |
8 | 7 | oveq2d 7367 | . 2 ⊢ (𝜑 → (𝑋 − (𝑌(+g‘𝐺)𝑍)) = (𝑋 − (𝑍(+g‘𝐺)𝑌))) |
9 | ablnncan.m | . . 3 ⊢ − = (-g‘𝐺) | |
10 | ablnncan.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 4, 5, 9, 1, 10, 2, 3 | ablsubsub4 19555 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌(+g‘𝐺)𝑍))) |
12 | 4, 5, 9, 1, 10, 3, 2 | ablsubsub4 19555 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑍) − 𝑌) = (𝑋 − (𝑍(+g‘𝐺)𝑌))) |
13 | 8, 11, 12 | 3eqtr4d 2787 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑍) − 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 Basecbs 17042 +gcplusg 17092 -gcsg 18709 Abelcabl 19521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-0g 17282 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-grp 18710 df-minusg 18711 df-sbg 18712 df-cmn 19522 df-abl 19523 |
This theorem is referenced by: ablnnncan1 19560 baerlem5alem2 40105 |
Copyright terms: Public domain | W3C validator |