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Mirrors > Home > MPE Home > Th. List > mgm2nsgrplem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for mgm2nsgrp 18081: 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 13753 | . . 3 ⊢ ((♯‘𝑆) = 2 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵)) |
3 | simp1 1132 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑆) | |
4 | simp2 1133 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆) | |
5 | 3, 3, 4 | 3jca 1124 | . . 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 1134 | . . . 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 2821 | . . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
11 | 1, 8, 9, 10 | mgm2nsgrplem2 18078 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = 𝐴) |
12 | 11 | 3adant3 1128 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = 𝐴) |
13 | 1, 8, 9, 10 | mgm2nsgrplem3 18079 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = 𝐵) |
14 | 13 | 3adant3 1128 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = 𝐵) |
15 | 7, 12, 14 | 3netr4d 3093 | . . 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 2830 | . . 3 ⊢ 𝑆 = (Base‘𝑀) |
18 | 17, 10 | isnsgrp 17899 | . 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 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∉ wnel 3123 ifcif 4467 {cpr 4563 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 2c2 11686 ♯chash 13684 Basecbs 16477 +gcplusg 16559 Smgrpcsgrp 17894 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 df-sgrp 17895 |
This theorem is referenced by: mgm2nsgrp 18081 mgmnsgrpex 18090 |
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