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Mirrors > Home > MPE Home > Th. List > ablnnncan1 | Structured version Visualization version GIF version |
Description: Cancellation law for group subtraction. (nnncan1 11521 analog.) (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 |
---|---|
ablnnncan1 | ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablnncan.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | ablnncan.m | . . 3 ⊢ − = (-g‘𝐺) | |
3 | ablnncan.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
4 | ablnncan.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | ablnncan.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | ablgrp 19734 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
7 | 3, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
8 | ablsub32.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
9 | 1, 2 | grpsubcl 18970 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 − 𝑍) ∈ 𝐵) |
10 | 7, 4, 8, 9 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑍) ∈ 𝐵) |
11 | 1, 2, 3, 4, 5, 10 | ablsub32 19770 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = ((𝑋 − (𝑋 − 𝑍)) − 𝑌)) |
12 | 1, 2, 3, 4, 8 | ablnncan 19769 | . . 3 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑍)) = 𝑍) |
13 | 12 | oveq1d 7430 | . 2 ⊢ (𝜑 → ((𝑋 − (𝑋 − 𝑍)) − 𝑌) = (𝑍 − 𝑌)) |
14 | 11, 13 | eqtrd 2768 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 Grpcgrp 18884 -gcsg 18886 Abelcabl 19730 |
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 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7988 df-2nd 7989 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-minusg 18888 df-sbg 18889 df-cmn 19731 df-abl 19732 |
This theorem is referenced by: minveclem2 25348 ply1divmo 26065 baerlem3lem2 41178 |
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