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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 7385. (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 18923 | . . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) |
9 | 2, 3, 5, 8 | syl3anc 1367 | . . . . . . 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 2856 | . . . . 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 3182 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))) |
16 | rinvmod.0 | . . 3 ⊢ 0 = (0g‘𝐺) | |
17 | cmnmnd 18922 | . . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
19 | 6, 16, 7, 18, 4 | mndinvmod 17941 | . 2 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) |
20 | rmoim 3731 | . 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 1537 ∈ wcel 2114 ∀wral 3138 ∃*wrmo 3141 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Mndcmnd 17911 CMndccmn 18906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-riota 7114 df-ov 7159 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-cmn 18908 |
This theorem is referenced by: (None) |
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