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Mirrors > Home > MPE Home > Th. List > sgrp2nmndlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for sgrp2nmnd 17240: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17077). (Contributed by AV, 29-Jan-2020.) |
Ref | Expression |
---|---|
mgm2nsgrp.s | ⊢ 𝑆 = {𝐴, 𝐵} |
mgm2nsgrp.b | ⊢ (Base‘𝑀) = 𝑆 |
sgrp2nmnd.o | ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)) |
Ref | Expression |
---|---|
sgrp2nmndlem1 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid1g 4239 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) | |
2 | mgm2nsgrp.s | . . 3 ⊢ 𝑆 = {𝐴, 𝐵} | |
3 | 1, 2 | syl6eleqr 2699 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆) |
4 | prid2g 4240 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵}) | |
5 | 4, 2 | syl6eleqr 2699 | . 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑆) |
6 | mgm2nsgrp.b | . . . 4 ⊢ (Base‘𝑀) = 𝑆 | |
7 | 6 | eqcomi 2619 | . . 3 ⊢ 𝑆 = (Base‘𝑀) |
8 | sgrp2nmnd.o | . . 3 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)) | |
9 | ne0i 3880 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅) | |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑆 ≠ ∅) |
11 | simpll 786 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐴 ∈ 𝑆) | |
12 | simplr 788 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐵 ∈ 𝑆) | |
13 | 7, 8, 10, 11, 12 | opifismgm 17081 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑀 ∈ Mgm) |
14 | 3, 5, 13 | syl2an 493 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 ifcif 4036 {cpr 4127 ‘cfv 5804 ↦ cmpt2 6551 Basecbs 15695 +gcplusg 15768 Mgmcmgm 17063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-mgm 17065 |
This theorem is referenced by: sgrp2nmndlem4 17238 |
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