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Mirrors > Home > MPE Home > Th. List > ablsubsub4 | 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 |
---|---|
ablsubsub4 | ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌 + 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablsubsub.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablgrp 19752 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . 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 18984 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) ∈ 𝐵) |
9 | 3, 4, 5, 8 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝐵) |
10 | ablsubsub.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
11 | ablsubadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
12 | eqid 2725 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
13 | 6, 11, 12, 7 | grpsubval 18950 | . . 3 ⊢ (((𝑋 − 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑌) + ((invg‘𝐺)‘𝑍))) |
14 | 9, 10, 13 | syl2anc 582 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = ((𝑋 − 𝑌) + ((invg‘𝐺)‘𝑍))) |
15 | 6, 12 | grpinvcl 18952 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵) |
16 | 3, 10, 15 | syl2anc 582 | . . 3 ⊢ (𝜑 → ((invg‘𝐺)‘𝑍) ∈ 𝐵) |
17 | 6, 11, 7, 1, 4, 5, 16 | ablsubsub 19784 | . 2 ⊢ (𝜑 → (𝑋 − (𝑌 − ((invg‘𝐺)‘𝑍))) = ((𝑋 − 𝑌) + ((invg‘𝐺)‘𝑍))) |
18 | 6, 11, 7, 12, 3, 5, 10 | grpsubinv 18976 | . . 3 ⊢ (𝜑 → (𝑌 − ((invg‘𝐺)‘𝑍)) = (𝑌 + 𝑍)) |
19 | 18 | oveq2d 7435 | . 2 ⊢ (𝜑 → (𝑋 − (𝑌 − ((invg‘𝐺)‘𝑍))) = (𝑋 − (𝑌 + 𝑍))) |
20 | 14, 17, 19 | 3eqtr2d 2771 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) − 𝑍) = (𝑋 − (𝑌 + 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 +gcplusg 17236 Grpcgrp 18898 invgcminusg 18899 -gcsg 18900 Abelcabl 19748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-sbg 18903 df-cmn 19749 df-abl 19750 |
This theorem is referenced by: ablsub32 19788 ablnnncan 19789 rngqiprngfulem4 21221 ip2subdi 21593 cpmadugsumlemF 22822 baerlem5alem2 41311 |
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