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Mirrors > Home > ILE Home > Th. List > ablsubsub4 | GIF version |
Description: Law for double subtraction. (Contributed by NM, 7-Apr-2015.) |
Ref | Expression |
---|---|
ablsubadd.b | ⊢ 𝐵 = (Base‘𝐺) |
ablsubadd.p | ⊢ + = (+g‘𝐺) |
ablsubadd.m | ⊢ − = (-g‘𝐺) |
ablsubsub.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablsubsub.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ablsubsub.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ablsubsub.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
ablsubsub4 | ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌 + 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablsubsub.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablgrp 12907 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | ablsubsub.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | ablsubsub.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | ablsubadd.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
7 | ablsubadd.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
8 | 6, 7 | grpsubcl 12826 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) ∈ 𝐵) |
9 | 3, 4, 5, 8 | syl3anc 1238 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝐵) |
10 | ablsubsub.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
11 | ablsubadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
12 | eqid 2177 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
13 | 6, 11, 12, 7 | grpsubval 12796 | . . 3 ⊢ (((𝑋 − 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑌) + ((invg‘𝐺)‘𝑍))) |
14 | 9, 10, 13 | syl2anc 411 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑌) + ((invg‘𝐺)‘𝑍))) |
15 | 6, 12 | grpinvcl 12798 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵) |
16 | 3, 10, 15 | syl2anc 411 | . . 3 ⊢ (𝜑 → ((invg‘𝐺)‘𝑍) ∈ 𝐵) |
17 | 6, 11, 7, 1, 4, 5, 16 | ablsubsub 12935 | . 2 ⊢ (𝜑 → (𝑋 − (𝑌 − ((invg‘𝐺)‘𝑍))) = ((𝑋 − 𝑌) + ((invg‘𝐺)‘𝑍))) |
18 | 6, 11, 7, 12, 3, 5, 10 | grpsubinv 12819 | . . 3 ⊢ (𝜑 → (𝑌 − ((invg‘𝐺)‘𝑍)) = (𝑌 + 𝑍)) |
19 | 18 | oveq2d 5884 | . 2 ⊢ (𝜑 → (𝑋 − (𝑌 − ((invg‘𝐺)‘𝑍))) = (𝑋 − (𝑌 + 𝑍))) |
20 | 14, 17, 19 | 3eqtr2d 2216 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌 + 𝑍))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 +gcplusg 12505 Grpcgrp 12754 invgcminusg 12755 -gcsg 12756 Abelcabl 12903 |
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 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-grp 12757 df-minusg 12758 df-sbg 12759 df-cmn 12904 df-abl 12905 |
This theorem is referenced by: ablsub32 12939 ablnnncan 12940 |
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