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Mirrors > Home > MPE Home > Th. List > simpgnideld | Structured version Visualization version GIF version |
Description: A simple group contains a nonidentity element. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
simpgnideld.1 | ⊢ 𝐵 = (Base‘𝐺) |
simpgnideld.2 | ⊢ 0 = (0g‘𝐺) |
simpgnideld.3 | ⊢ (𝜑 → 𝐺 ∈ SimpGrp) |
Ref | Expression |
---|---|
simpgnideld | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ¬ 𝑥 = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpgnideld.1 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | simpgnideld.2 | . . . 4 ⊢ 0 = (0g‘𝐺) | |
3 | simpgnideld.3 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) | |
4 | 1, 2, 3 | simpgntrivd 20054 | . . 3 ⊢ (𝜑 → ¬ 𝐵 = { 0 }) |
5 | 3 | simpggrpd 20051 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp) |
6 | grpmnd 18896 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
7 | 1, 2 | mndidcl 18703 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
8 | 5, 6, 7 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵) |
9 | 8 | ne0d 4332 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅) |
10 | eqsn 4829 | . . . 4 ⊢ (𝐵 ≠ ∅ → (𝐵 = { 0 } ↔ ∀𝑥 ∈ 𝐵 𝑥 = 0 )) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝐵 = { 0 } ↔ ∀𝑥 ∈ 𝐵 𝑥 = 0 )) |
12 | 4, 11 | mtbid 323 | . 2 ⊢ (𝜑 → ¬ ∀𝑥 ∈ 𝐵 𝑥 = 0 ) |
13 | rexnal 3090 | . 2 ⊢ (∃𝑥 ∈ 𝐵 ¬ 𝑥 = 0 ↔ ¬ ∀𝑥 ∈ 𝐵 𝑥 = 0 ) | |
14 | 12, 13 | sylibr 233 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ¬ 𝑥 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ∅c0 4319 {csn 4625 ‘cfv 6543 Basecbs 17174 0gc0g 17415 Mndcmnd 18688 Grpcgrp 18889 SimpGrpcsimpg 20046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-nsg 19078 df-simpg 20047 |
This theorem is referenced by: ablsimpgcygd 20062 ablsimpgfind 20066 |
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