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Mirrors > Home > MPE Home > Th. List > mgm2nsgrplem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for mgm2nsgrp 18025: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17853). (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 4688 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) | |
2 | mgm2nsgrp.s | . . 3 ⊢ 𝑆 = {𝐴, 𝐵} | |
3 | 1, 2 | eleqtrrdi 2921 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆) |
4 | prid2g 4689 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵}) | |
5 | 4, 2 | eleqtrrdi 2921 | . 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑆) |
6 | mgm2nsgrp.b | . . . 4 ⊢ (Base‘𝑀) = 𝑆 | |
7 | 6 | eqcomi 2827 | . . 3 ⊢ 𝑆 = (Base‘𝑀) |
8 | mgm2nsgrp.o | . . 3 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)) | |
9 | ne0i 4297 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅) | |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑆 ≠ ∅) |
11 | simplr 765 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐵 ∈ 𝑆) | |
12 | simpll 763 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐴 ∈ 𝑆) | |
13 | 7, 8, 10, 11, 12 | opifismgm 17857 | . 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 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 ifcif 4463 {cpr 4559 ‘cfv 6348 ∈ cmpo 7147 Basecbs 16471 +gcplusg 16553 Mgmcmgm 17838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-mgm 17840 |
This theorem is referenced by: mgm2nsgrp 18025 mgmnsgrpex 18034 |
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