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Mirrors > Home > MPE Home > Th. List > sgrp2nmndlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for sgrp2nmnd 18956: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 18681). (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 4765 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) | |
2 | mgm2nsgrp.s | . . 3 ⊢ 𝑆 = {𝐴, 𝐵} | |
3 | 1, 2 | eleqtrrdi 2850 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆) |
4 | prid2g 4766 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵}) | |
5 | 4, 2 | eleqtrrdi 2850 | . 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑆) |
6 | mgm2nsgrp.b | . . . 4 ⊢ (Base‘𝑀) = 𝑆 | |
7 | 6 | eqcomi 2744 | . . 3 ⊢ 𝑆 = (Base‘𝑀) |
8 | sgrp2nmnd.o | . . 3 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)) | |
9 | ne0i 4347 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅) | |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑆 ≠ ∅) |
11 | simpll 767 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐴 ∈ 𝑆) | |
12 | simplr 769 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐵 ∈ 𝑆) | |
13 | 7, 8, 10, 11, 12 | opifismgm 18685 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑀 ∈ Mgm) |
14 | 3, 5, 13 | syl2an 596 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 ifcif 4531 {cpr 4633 ‘cfv 6563 ∈ cmpo 7433 Basecbs 17245 +gcplusg 17298 Mgmcmgm 18664 |
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 ax-un 7754 |
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-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 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-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-mgm 18666 |
This theorem is referenced by: sgrp2nmndlem4 18954 |
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