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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 7670. (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 19831 | . . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝑤 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝑤 + 𝐴) = (𝐴 + 𝑤)) |
9 | 2, 3, 5, 8 | syl3anc 1370 | . . . . . . 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 2775 | . . . . 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 3144 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ((𝐴 + 𝑤) = 0 → ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))) |
16 | rinvmod.0 | . . 3 ⊢ 0 = (0g‘𝐺) | |
17 | cmnmnd 19830 | . . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
19 | 6, 16, 7, 18, 4 | mndinvmod 18790 | . 2 ⊢ (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 )) |
20 | rmoim 3749 | . 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 1537 ∈ wcel 2106 ∀wral 3059 ∃*wrmo 3377 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 Mndcmnd 18760 CMndccmn 19813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-cmn 19815 |
This theorem is referenced by: (None) |
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