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Mirrors > Home > MPE Home > Th. List > mgm2nsgrplem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for mgm2nsgrp 18196: M is not a semigroup. (Contributed by AV, 28-Jan-2020.) (Proof shortened by AV, 31-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 |
---|---|
mgm2nsgrplem4 | ⊢ ((♯‘𝑆) = 2 → 𝑀 ∉ Smgrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgm2nsgrp.s | . . . 4 ⊢ 𝑆 = {𝐴, 𝐵} | |
2 | 1 | hashprdifel 13844 | . . 3 ⊢ ((♯‘𝑆) = 2 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵)) |
3 | simp1 1137 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑆) | |
4 | simp2 1138 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆) | |
5 | 3, 3, 4 | 3jca 1129 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
6 | 2, 5 | syl 17 | . 2 ⊢ ((♯‘𝑆) = 2 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
7 | simp3 1139 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵) | |
8 | mgm2nsgrp.b | . . . . . 6 ⊢ (Base‘𝑀) = 𝑆 | |
9 | mgm2nsgrp.o | . . . . . 6 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if((𝑥 = 𝐴 ∧ 𝑦 = 𝐴), 𝐵, 𝐴)) | |
10 | eqid 2738 | . . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
11 | 1, 8, 9, 10 | mgm2nsgrplem2 18193 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = 𝐴) |
12 | 11 | 3adant3 1133 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = 𝐴) |
13 | 1, 8, 9, 10 | mgm2nsgrplem3 18194 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = 𝐵) |
14 | 13 | 3adant3 1133 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = 𝐵) |
15 | 7, 12, 14 | 3netr4d 3011 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) ≠ (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) |
16 | 2, 15 | syl 17 | . 2 ⊢ ((♯‘𝑆) = 2 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) ≠ (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) |
17 | 8 | eqcomi 2747 | . . 3 ⊢ 𝑆 = (Base‘𝑀) |
18 | 17, 10 | isnsgrp 18014 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) ≠ (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) → 𝑀 ∉ Smgrp)) |
19 | 6, 16, 18 | sylc 65 | 1 ⊢ ((♯‘𝑆) = 2 → 𝑀 ∉ Smgrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∉ wnel 3038 ifcif 4411 {cpr 4515 ‘cfv 6333 (class class class)co 7164 ∈ cmpo 7166 2c2 11764 ♯chash 13775 Basecbs 16579 +gcplusg 16661 Smgrpcsgrp 18009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-oadd 8128 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-dju 9396 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-hash 13776 df-sgrp 18010 |
This theorem is referenced by: mgm2nsgrp 18196 mgmnsgrpex 18205 |
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