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Mirrors > Home > MPE Home > Th. List > ablnncan | Structured version Visualization version GIF version |
Description: Cancellation law for group subtraction. (nncan 11496 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 2731 | . . 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 19733 | . 2 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = ((𝑋 − 𝑋)(+g‘𝐺)𝑌)) |
8 | ablgrp 19701 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
9 | 4, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
10 | eqid 2731 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
11 | 1, 10, 3 | grpsubid 18950 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺)) |
12 | 9, 5, 11 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺)) |
13 | 12 | oveq1d 7427 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑋)(+g‘𝐺)𝑌) = ((0g‘𝐺)(+g‘𝐺)𝑌)) |
14 | 1, 2, 10 | grplid 18895 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺)(+g‘𝐺)𝑌) = 𝑌) |
15 | 9, 6, 14 | syl2anc 583 | . 2 ⊢ (𝜑 → ((0g‘𝐺)(+g‘𝐺)𝑌) = 𝑌) |
16 | 7, 13, 15 | 3eqtrd 2775 | 1 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑌)) = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 0gc0g 17392 Grpcgrp 18861 -gcsg 18863 Abelcabl 19697 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-sbg 18866 df-cmn 19698 df-abl 19699 |
This theorem is referenced by: ablnnncan1 19739 pgpfac1lem3 19995 rngqiprngfulem4 21162 tsmsxplem1 23977 baerlem5blem2 41047 |
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