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Mirrors > Home > MPE Home > Th. List > ablpnpcan | Structured version Visualization version GIF version |
Description: Cancellation law for mixed addition and subtraction. (pnpcan 10522 analog.) (Contributed by NM, 29-May-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) |
ablpnpcan.g | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablpnpcan.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ablpnpcan.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ablpnpcan.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
ablpnpcan | ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablsubsub.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablsubsub.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | ablsubsub.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | ablsubsub.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
5 | ablsubadd.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
6 | ablsubadd.p | . . . 4 ⊢ + = (+g‘𝐺) | |
7 | ablsubadd.m | . . . 4 ⊢ − = (-g‘𝐺) | |
8 | 5, 6, 7 | ablsub4 18425 | . . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = ((𝑋 − 𝑋) + (𝑌 − 𝑍))) |
9 | 1, 2, 3, 2, 4, 8 | syl122anc 1485 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = ((𝑋 − 𝑋) + (𝑌 − 𝑍))) |
10 | ablgrp 18405 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
11 | 1, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
12 | eqid 2771 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
13 | 5, 12, 7 | grpsubid 17707 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 − 𝑋) = (0g‘𝐺)) |
14 | 11, 2, 13 | syl2anc 573 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐺)) |
15 | 14 | oveq1d 6808 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑋) + (𝑌 − 𝑍)) = ((0g‘𝐺) + (𝑌 − 𝑍))) |
16 | 5, 7 | grpsubcl 17703 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵) |
17 | 11, 3, 4, 16 | syl3anc 1476 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑍) ∈ 𝐵) |
18 | 5, 6, 12 | grplid 17660 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 − 𝑍) ∈ 𝐵) → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍)) |
19 | 11, 17, 18 | syl2anc 573 | . 2 ⊢ (𝜑 → ((0g‘𝐺) + (𝑌 − 𝑍)) = (𝑌 − 𝑍)) |
20 | 9, 15, 19 | 3eqtrd 2809 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) − (𝑋 + 𝑍)) = (𝑌 − 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ‘cfv 6031 (class class class)co 6793 Basecbs 16064 +gcplusg 16149 0gc0g 16308 Grpcgrp 17630 -gcsg 17632 Abelcabl 18401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-1st 7315 df-2nd 7316 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-minusg 17634 df-sbg 17635 df-cmn 18402 df-abl 18403 |
This theorem is referenced by: hdmaprnlem7N 37665 |
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