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Mirrors > Home > MPE Home > Th. List > mgm2nsgrplem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for mgm2nsgrp 18090: 𝑀 is a magma, even if 𝐴 = 𝐵 (𝑀 is the trivial magma in this case, see mgmb1mgm1 17868). (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 4699 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) | |
2 | mgm2nsgrp.s | . . 3 ⊢ 𝑆 = {𝐴, 𝐵} | |
3 | 1, 2 | eleqtrrdi 2927 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑆) |
4 | prid2g 4700 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵}) | |
5 | 4, 2 | eleqtrrdi 2927 | . 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑆) |
6 | mgm2nsgrp.b | . . . 4 ⊢ (Base‘𝑀) = 𝑆 | |
7 | 6 | eqcomi 2833 | . . 3 ⊢ 𝑆 = (Base‘𝑀) |
8 | mgm2nsgrp.o | . . 3 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)) | |
9 | ne0i 4303 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅) | |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑆 ≠ ∅) |
11 | simplr 767 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐵 ∈ 𝑆) | |
12 | simpll 765 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐴 ∈ 𝑆) | |
13 | 7, 8, 10, 11, 12 | opifismgm 17872 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑀 ∈ Mgm) |
14 | 3, 5, 13 | syl2an 597 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∅c0 4294 ifcif 4470 {cpr 4572 ‘cfv 6358 ∈ cmpo 7161 Basecbs 16486 +gcplusg 16568 Mgmcmgm 17853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-mgm 17855 |
This theorem is referenced by: mgm2nsgrp 18090 mgmnsgrpex 18099 |
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