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Mirrors > Home > MPE Home > Th. List > mgm2nsgrplem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for mgm2nsgrp 18865: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 18606). (Contributed by AV, 27-Jan-2020.) |
Ref | Expression |
---|---|
mgm2nsgrp.s | ⊢ 𝑆 = {𝐴, 𝐵} |
mgm2nsgrp.b | ⊢ (Base‘𝑀) = 𝑆 |
mgm2nsgrp.o | ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)) |
Ref | Expression |
---|---|
mgm2nsgrplem1 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid1g 4760 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) | |
2 | mgm2nsgrp.s | . . 3 ⊢ 𝑆 = {𝐴, 𝐵} | |
3 | 1, 2 | eleqtrrdi 2839 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆) |
4 | prid2g 4761 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵}) | |
5 | 4, 2 | eleqtrrdi 2839 | . 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑆) |
6 | mgm2nsgrp.b | . . . 4 ⊢ (Base‘𝑀) = 𝑆 | |
7 | 6 | eqcomi 2736 | . . 3 ⊢ 𝑆 = (Base‘𝑀) |
8 | mgm2nsgrp.o | . . 3 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)) | |
9 | ne0i 4330 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅) | |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑆 ≠ ∅) |
11 | simplr 768 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐵 ∈ 𝑆) | |
12 | simpll 766 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐴 ∈ 𝑆) | |
13 | 7, 8, 10, 11, 12 | opifismgm 18610 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑀 ∈ Mgm) |
14 | 3, 5, 13 | syl2an 595 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∅c0 4318 ifcif 4524 {cpr 4626 ‘cfv 6542 ∈ cmpo 7416 Basecbs 17171 +gcplusg 17224 Mgmcmgm 18589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-mgm 18591 |
This theorem is referenced by: mgm2nsgrp 18865 mgmnsgrpex 18874 |
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