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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 2825 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 4, 5 | ablcom 18570 | . . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)𝑌)) |
7 | 1, 2, 3, 6 | syl3anc 1494 | . . 3 ⊢ (𝜑 → (𝑌(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)𝑌)) |
8 | 7 | oveq2d 6926 | . 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 18584 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌(+g‘𝐺)𝑍))) |
12 | 4, 5, 9, 1, 10, 3, 2 | ablsubsub4 18584 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑍) − 𝑌) = (𝑋 − (𝑍(+g‘𝐺)𝑌))) |
13 | 8, 11, 12 | 3eqtr4d 2871 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑍) − 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 -gcsg 17785 Abelcabl 18554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-sbg 17788 df-cmn 18555 df-abl 18556 |
This theorem is referenced by: ablnnncan1 18589 baerlem5alem2 37781 |
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