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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 2726 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 4, 5 | ablcom 19797 | . . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)𝑌)) |
7 | 1, 2, 3, 6 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑌(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)𝑌)) |
8 | 7 | oveq2d 7440 | . 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 19816 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌(+g‘𝐺)𝑍))) |
12 | 4, 5, 9, 1, 10, 3, 2 | ablsubsub4 19816 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑍) − 𝑌) = (𝑋 − (𝑍(+g‘𝐺)𝑌))) |
13 | 8, 11, 12 | 3eqtr4d 2776 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑍) − 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 +gcplusg 17266 -gcsg 18930 Abelcabl 19779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-minusg 18932 df-sbg 18933 df-cmn 19780 df-abl 19781 |
This theorem is referenced by: ablnnncan1 19821 baerlem5alem2 41410 |
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