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Mirrors > Home > MPE Home > Th. List > ablsubsub | Structured version Visualization version 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 |
---|---|
ablsubsub | ⊢ (𝜑 → (𝑋 − (𝑌 − 𝑍)) = ((𝑋 − 𝑌) + 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablsubsub.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablgrp 18550 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | ablsubsub.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | ablsubsub.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | ablsubsub.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
7 | ablsubadd.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
8 | ablsubadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
9 | ablsubadd.m | . . . 4 ⊢ − = (-g‘𝐺) | |
10 | 7, 8, 9 | grpsubsub 17857 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 − (𝑌 − 𝑍)) = (𝑋 + (𝑍 − 𝑌))) |
11 | 3, 4, 5, 6, 10 | syl13anc 1497 | . 2 ⊢ (𝜑 → (𝑋 − (𝑌 − 𝑍)) = (𝑋 + (𝑍 − 𝑌))) |
12 | 7, 8, 9 | grpaddsubass 17858 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + 𝑍) − 𝑌) = (𝑋 + (𝑍 − 𝑌))) |
13 | 3, 4, 6, 5, 12 | syl13anc 1497 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑍) − 𝑌) = (𝑋 + (𝑍 − 𝑌))) |
14 | 7, 8, 9 | abladdsub 18572 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + 𝑍) − 𝑌) = ((𝑋 − 𝑌) + 𝑍)) |
15 | 1, 4, 6, 5, 14 | syl13anc 1497 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑍) − 𝑌) = ((𝑋 − 𝑌) + 𝑍)) |
16 | 11, 13, 15 | 3eqtr2d 2866 | 1 ⊢ (𝜑 → (𝑋 − (𝑌 − 𝑍)) = ((𝑋 − 𝑌) + 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ‘cfv 6122 (class class class)co 6904 Basecbs 16221 +gcplusg 16304 Grpcgrp 17775 -gcsg 17777 Abelcabl 18546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-1st 7427 df-2nd 7428 df-0g 16454 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-grp 17778 df-minusg 17779 df-sbg 17780 df-cmn 18547 df-abl 18548 |
This theorem is referenced by: ablsubsub4 18576 ablnncan 18578 ip2subdi 20350 |
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