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Mirrors > Home > MPE Home > Th. List > ablnnncan | Structured version Visualization version GIF version |
Description: Cancellation law for group subtraction. (nnncan 11432 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.) |
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 |
---|---|
ablnnncan | ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablnncan.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2736 | . . 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 | ablgrp 19558 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
8 | ablnncan.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | ablsub32.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
10 | 1, 3 | grpsubcl 18818 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵) |
11 | 7, 8, 9, 10 | syl3anc 1371 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑍) ∈ 𝐵) |
12 | 1, 2, 3, 4, 5, 11, 9 | ablsubsub4 19588 | . 2 ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − ((𝑌 − 𝑍)(+g‘𝐺)𝑍))) |
13 | 1, 2 | ablcom 19572 | . . . . 5 ⊢ ((𝐺 ∈ Abel ∧ (𝑌 − 𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑌 − 𝑍)(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)(𝑌 − 𝑍))) |
14 | 4, 11, 9, 13 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → ((𝑌 − 𝑍)(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)(𝑌 − 𝑍))) |
15 | 1, 2, 3 | ablpncan3 19586 | . . . . 5 ⊢ ((𝐺 ∈ Abel ∧ (𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑍(+g‘𝐺)(𝑌 − 𝑍)) = 𝑌) |
16 | 4, 9, 8, 15 | syl12anc 835 | . . . 4 ⊢ (𝜑 → (𝑍(+g‘𝐺)(𝑌 − 𝑍)) = 𝑌) |
17 | 14, 16 | eqtrd 2776 | . . 3 ⊢ (𝜑 → ((𝑌 − 𝑍)(+g‘𝐺)𝑍) = 𝑌) |
18 | 17 | oveq2d 7369 | . 2 ⊢ (𝜑 → (𝑋 − ((𝑌 − 𝑍)(+g‘𝐺)𝑍)) = (𝑋 − 𝑌)) |
19 | 12, 18 | eqtrd 2776 | 1 ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 +gcplusg 17125 Grpcgrp 18740 -gcsg 18742 Abelcabl 19554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-0g 17315 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-grp 18743 df-minusg 18744 df-sbg 18745 df-cmn 19555 df-abl 19556 |
This theorem is referenced by: (None) |
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