Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rinvmod | Structured version Visualization version GIF version |
Description: Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovmo 7382. (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 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝐺 ∈ CMnd) |
3 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵) | |
4 | rinvmod.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
5 | 4 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝐴 ∈ 𝐵) |
6 | rinvmod.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺) | |
7 | rinvmod.p | . . . . . . . . 9 ⊢ + = (+g‘𝐺) | |
8 | 6, 7 | cmncom 18919 | . . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) |
9 | 2, 3, 5, 8 | syl3anc 1366 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) |
10 | 9 | adantr 483 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) |
11 | simpr 487 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝐴 + 𝑤) = 0 ) | |
12 | 10, 11 | eqtrd 2855 | . . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = 0 ) |
13 | 12, 11 | jca 514 | . . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) |
14 | 13 | ex 415 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))) |
15 | 14 | ralrimiva 3181 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))) |
16 | rinvmod.0 | . . 3 ⊢ 0 = (0g‘𝐺) | |
17 | cmnmnd 18918 | . . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
19 | 6, 16, 7, 18, 4 | mndinvmod 17937 | . 2 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) |
20 | rmoim 3729 | . 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 65 | 1 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∃*wrmo 3140 ‘cfv 6352 (class class class)co 7153 Basecbs 16479 +gcplusg 16561 0gc0g 16709 Mndcmnd 17907 CMndccmn 18902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-mpt 5144 df-id 5457 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-iota 6311 df-fun 6354 df-fv 6360 df-riota 7111 df-ov 7156 df-0g 16711 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-cmn 18904 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |