![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ablnnncan | GIF version |
Description: Cancellation law for group subtraction. (nnncan 8205 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.) |
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 |
---|---|
ablnnncan | ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablnncan.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2187 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | ablnncan.m | . . 3 ⊢ − = (-g‘𝐺) | |
4 | ablnncan.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
5 | ablnncan.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | ablgrp 13123 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
7 | 4, 6 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
8 | ablnncan.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | ablsub32.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
10 | 1, 3 | grpsubcl 12974 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) ∈ 𝐵) |
11 | 7, 8, 9, 10 | syl3anc 1248 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑍) ∈ 𝐵) |
12 | 1, 2, 3, 4, 5, 11, 9 | ablsubsub4 13153 | . 2 ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − ((𝑌 − 𝑍)(+g‘𝐺)𝑍))) |
13 | 1, 2 | ablcom 13137 | . . . . 5 ⊢ ((𝐺 ∈ Abel ∧ (𝑌 − 𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑌 − 𝑍)(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)(𝑌 − 𝑍))) |
14 | 4, 11, 9, 13 | syl3anc 1248 | . . . 4 ⊢ (𝜑 → ((𝑌 − 𝑍)(+g‘𝐺)𝑍) = (𝑍(+g‘𝐺)(𝑌 − 𝑍))) |
15 | 1, 2, 3 | ablpncan3 13151 | . . . . 5 ⊢ ((𝐺 ∈ Abel ∧ (𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑍(+g‘𝐺)(𝑌 − 𝑍)) = 𝑌) |
16 | 4, 9, 8, 15 | syl12anc 1246 | . . . 4 ⊢ (𝜑 → (𝑍(+g‘𝐺)(𝑌 − 𝑍)) = 𝑌) |
17 | 14, 16 | eqtrd 2220 | . . 3 ⊢ (𝜑 → ((𝑌 − 𝑍)(+g‘𝐺)𝑍) = 𝑌) |
18 | 17 | oveq2d 5904 | . 2 ⊢ (𝜑 → (𝑋 − ((𝑌 − 𝑍)(+g‘𝐺)𝑍)) = (𝑋 − 𝑌)) |
19 | 12, 18 | eqtrd 2220 | 1 ⊢ (𝜑 → ((𝑋 − (𝑌 − 𝑍)) − 𝑍) = (𝑋 − 𝑌)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 ‘cfv 5228 (class class class)co 5888 Basecbs 12475 +gcplusg 12550 Grpcgrp 12896 -gcsg 12898 Abelcabl 13119 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-inn 8933 df-2 8991 df-ndx 12478 df-slot 12479 df-base 12481 df-plusg 12563 df-0g 12724 df-mgm 12793 df-sgrp 12826 df-mnd 12837 df-grp 12899 df-minusg 12900 df-sbg 12901 df-cmn 13120 df-abl 13121 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |