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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 2821 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 4, 5 | ablcom 18924 | . . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)𝑌)) |
7 | 1, 2, 3, 6 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑌(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)𝑌)) |
8 | 7 | oveq2d 7172 | . 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 18939 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌(+g‘𝐺)𝑍))) |
12 | 4, 5, 9, 1, 10, 3, 2 | ablsubsub4 18939 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑍) − 𝑌) = (𝑋 − (𝑍(+g‘𝐺)𝑌))) |
13 | 8, 11, 12 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑍) − 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 -gcsg 18105 Abelcabl 18907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-cmn 18908 df-abl 18909 |
This theorem is referenced by: ablnnncan1 18944 baerlem5alem2 38862 |
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