![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rinvmod | GIF version |
Description: Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6061. (Contributed by AV, 31-Dec-2023.) |
Ref | Expression |
---|---|
rinvmod.b | ⊢ 𝐵 = (Base‘𝐺) |
rinvmod.0 | ⊢ 0 = (0g‘𝐺) |
rinvmod.p | ⊢ + = (+g‘𝐺) |
rinvmod.m | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
rinvmod.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
Ref | Expression |
---|---|
rinvmod | ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rinvmod.m | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
2 | 1 | adantr 276 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝐺 ∈ CMnd) |
3 | simpr 110 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵) | |
4 | rinvmod.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
5 | 4 | adantr 276 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝐴 ∈ 𝐵) |
6 | rinvmod.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺) | |
7 | rinvmod.p | . . . . . . . . 9 ⊢ + = (+g‘𝐺) | |
8 | 6, 7 | cmncom 12919 | . . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) |
9 | 2, 3, 5, 8 | syl3anc 1238 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) |
10 | 9 | adantr 276 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) |
11 | simpr 110 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝐴 + 𝑤) = 0 ) | |
12 | 10, 11 | eqtrd 2210 | . . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = 0 ) |
13 | 12, 11 | jca 306 | . . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) |
14 | 13 | ex 115 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))) |
15 | 14 | ralrimiva 2550 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))) |
16 | rinvmod.0 | . . 3 ⊢ 0 = (0g‘𝐺) | |
17 | cmnmnd 12918 | . . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
18 | 1, 17 | syl 14 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
19 | 6, 16, 7, 18, 4 | mndinvmod 12723 | . 2 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) |
20 | rmoim 2938 | . 2 ⊢ (∀𝑤 ∈ 𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) → (∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ) → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 )) | |
21 | 15, 19, 20 | sylc 62 | 1 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃*wrmo 2458 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 +gcplusg 12505 0gc0g 12640 Mndcmnd 12696 CMndccmn 12902 |
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-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-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 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-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 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-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-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-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-riota 5824 df-ov 5871 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-cmn 12904 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |