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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 7671. (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 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝐺 ∈ CMnd) | 
| 3 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵) | |
| 4 | rinvmod.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
| 5 | 4 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → 𝐴 ∈ 𝐵) | 
| 6 | rinvmod.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺) | |
| 7 | rinvmod.p | . . . . . . . . 9 ⊢ + = (+g‘𝐺) | |
| 8 | 6, 7 | cmncom 19817 | . . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) | 
| 9 | 2, 3, 5, 8 | syl3anc 1372 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) | 
| 10 | 9 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) | 
| 11 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝐴 + 𝑤) = 0 ) | |
| 12 | 10, 11 | eqtrd 2776 | . . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → (𝑤 + 𝐴) = 0 ) | 
| 13 | 12, 11 | jca 511 | . . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐵) ∧ (𝐴 + 𝑤) = 0 ) → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) | 
| 14 | 13 | ex 412 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐵) → ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))) | 
| 15 | 14 | ralrimiva 3145 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))) | 
| 16 | rinvmod.0 | . . 3 ⊢ 0 = (0g‘𝐺) | |
| 17 | cmnmnd 19816 | . . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
| 18 | 1, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) | 
| 19 | 6, 16, 7, 18, 4 | mndinvmod 18778 | . 2 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) | 
| 20 | rmoim 3745 | . 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 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃*wrmo 3378 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Mndcmnd 18748 CMndccmn 19799 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-riota 7389 df-ov 7435 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-cmn 19801 | 
| This theorem is referenced by: (None) | 
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