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Mirrors > Home > ILE Home > Th. List > ablnnncan1 | GIF version |
Description: Cancellation law for group subtraction. (nnncan1 8191 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 13046 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
7 | 3, 6 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
8 | ablsub32.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
9 | 1, 2 | grpsubcl 12904 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 − 𝑍) ∈ 𝐵) |
10 | 7, 4, 8, 9 | syl3anc 1238 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑍) ∈ 𝐵) |
11 | 1, 2, 3, 4, 5, 10 | ablsub32 13078 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = ((𝑋 − (𝑋 − 𝑍)) − 𝑌)) |
12 | 1, 2, 3, 4, 8 | ablnncan 13077 | . . 3 ⊢ (𝜑 → (𝑋 − (𝑋 − 𝑍)) = 𝑍) |
13 | 12 | oveq1d 5889 | . 2 ⊢ (𝜑 → ((𝑋 − (𝑋 − 𝑍)) − 𝑌) = (𝑍 − 𝑌)) |
14 | 11, 13 | eqtrd 2210 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − (𝑋 − 𝑍)) = (𝑍 − 𝑌)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 Grpcgrp 12831 -gcsg 12833 Abelcabl 13042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-inn 8918 df-2 8976 df-ndx 12459 df-slot 12460 df-base 12462 df-plusg 12543 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-grp 12834 df-minusg 12835 df-sbg 12836 df-cmn 13043 df-abl 13044 |
This theorem is referenced by: (None) |
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