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Mirrors > Home > MPE Home > Th. List > ablnnncan1 | Structured version Visualization version GIF version |
Description: Cancellation law for group subtraction. (nnncan1 10908 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 18894 | . . . . 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 18162 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 − 𝑍) ∈ 𝐵) |
10 | 7, 4, 8, 9 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑍) ∈ 𝐵) |
11 | 1, 2, 3, 4, 5, 10 | ablsub32 18925 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = ((𝑋 − (𝑋 − 𝑍)) − 𝑌)) |
12 | 1, 2, 3, 4, 8 | ablnncan 18924 | . . 3 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑍)) = 𝑍) |
13 | 12 | oveq1d 7157 | . 2 ⊢ (𝜑 → ((𝑋 − (𝑋 − 𝑍)) − 𝑌) = (𝑍 − 𝑌)) |
14 | 11, 13 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 Grpcgrp 18086 -gcsg 18088 Abelcabl 18890 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-1st 7675 df-2nd 7676 df-0g 16698 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-grp 18089 df-minusg 18090 df-sbg 18091 df-cmn 18891 df-abl 18892 |
This theorem is referenced by: minveclem2 24012 ply1divmo 24715 baerlem3lem2 38878 |
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