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Mirrors > Home > MPE Home > Th. List > ablnncan | Structured version Visualization version GIF version |
Description: Cancellation law for group subtraction. (nncan 10189 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) |
Ref | Expression |
---|---|
ablnncan | ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablnncan.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2610 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | ablnncan.m | . . 3 ⊢ − = (-g‘𝐺) | |
4 | ablnncan.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
5 | ablnncan.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | ablnncan.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | 1, 2, 3, 4, 5, 5, 6 | ablsubsub 18046 | . 2 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = ((𝑋 − 𝑋)(+g‘𝐺)𝑌)) |
8 | ablgrp 18021 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
9 | 4, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
10 | eqid 2610 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
11 | 1, 10, 3 | grpsubid 17322 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺)) |
12 | 9, 5, 11 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺)) |
13 | 12 | oveq1d 6564 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑋)(+g‘𝐺)𝑌) = ((0g‘𝐺)(+g‘𝐺)𝑌)) |
14 | 1, 2, 10 | grplid 17275 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺)(+g‘𝐺)𝑌) = 𝑌) |
15 | 9, 6, 14 | syl2anc 691 | . 2 ⊢ (𝜑 → ((0g‘𝐺)(+g‘𝐺)𝑌) = 𝑌) |
16 | 7, 13, 15 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 -gcsg 17247 Abelcabl 18017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-cmn 18018 df-abl 18019 |
This theorem is referenced by: ablnnncan1 18052 pgpfac1lem3 18299 tsmsxplem1 21766 baerlem5blem2 36019 |
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