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Mirrors > Home > MPE Home > Th. List > ablsimpgfindlem2 | Structured version Visualization version GIF version |
Description: Lemma for ablsimpgfind 19494. An element of an abelian finite simple group which squares to the identity has finite order. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
ablsimpgfindlem1.1 | ⊢ 𝐵 = (Base‘𝐺) |
ablsimpgfindlem1.2 | ⊢ 0 = (0g‘𝐺) |
ablsimpgfindlem1.3 | ⊢ · = (.g‘𝐺) |
ablsimpgfindlem1.4 | ⊢ 𝑂 = (od‘𝐺) |
ablsimpgfindlem1.5 | ⊢ (𝜑 → 𝐺 ∈ Abel) |
ablsimpgfindlem1.6 | ⊢ (𝜑 → 𝐺 ∈ SimpGrp) |
Ref | Expression |
---|---|
ablsimpgfindlem2 | ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → (𝑂‘𝑥) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → (2 · 𝑥) = 0 ) | |
2 | ablsimpgfindlem1.6 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) | |
3 | 2 | simpggrpd 19479 | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐺 ∈ Grp) |
5 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
6 | 2z 12206 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 2 ∈ ℤ) |
8 | 4, 5, 7 | 3jca 1130 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 2 ∈ ℤ)) |
9 | 8 | adantr 484 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → (𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 2 ∈ ℤ)) |
10 | ablsimpgfindlem1.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
11 | ablsimpgfindlem1.4 | . . . . 5 ⊢ 𝑂 = (od‘𝐺) | |
12 | ablsimpgfindlem1.3 | . . . . 5 ⊢ · = (.g‘𝐺) | |
13 | ablsimpgfindlem1.2 | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
14 | 10, 11, 12, 13 | oddvds 18936 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 2 ∈ ℤ) → ((𝑂‘𝑥) ∥ 2 ↔ (2 · 𝑥) = 0 )) |
15 | 9, 14 | syl 17 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → ((𝑂‘𝑥) ∥ 2 ↔ (2 · 𝑥) = 0 )) |
16 | 1, 15 | mpbird 260 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → (𝑂‘𝑥) ∥ 2) |
17 | 2ne0 11931 | . . . . 5 ⊢ 2 ≠ 0 | |
18 | 17 | a1i 11 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → 2 ≠ 0) |
19 | 18 | neneqd 2944 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → ¬ 2 = 0) |
20 | 0dvds 15835 | . . . 4 ⊢ (2 ∈ ℤ → (0 ∥ 2 ↔ 2 = 0)) | |
21 | 6, 20 | ax-mp 5 | . . 3 ⊢ (0 ∥ 2 ↔ 2 = 0) |
22 | 19, 21 | sylnibr 332 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → ¬ 0 ∥ 2) |
23 | nbrne2 5070 | . 2 ⊢ (((𝑂‘𝑥) ∥ 2 ∧ ¬ 0 ∥ 2) → (𝑂‘𝑥) ≠ 0) | |
24 | 16, 22, 23 | syl2anc 587 | 1 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → (𝑂‘𝑥) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2939 class class class wbr 5050 ‘cfv 6377 (class class class)co 7210 0cc0 10726 2c2 11882 ℤcz 12173 ∥ cdvds 15812 Basecbs 16757 0gc0g 16941 Grpcgrp 18362 .gcmg 18485 odcod 18913 Abelcabl 19168 SimpGrpcsimpg 19474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 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-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-sup 9055 df-inf 9056 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-n0 12088 df-z 12174 df-uz 12436 df-rp 12584 df-fz 13093 df-fl 13364 df-mod 13440 df-seq 13572 df-exp 13633 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 df-dvds 15813 df-0g 16943 df-mgm 18111 df-sgrp 18160 df-mnd 18171 df-grp 18365 df-minusg 18366 df-sbg 18367 df-mulg 18486 df-od 18917 df-simpg 19475 |
This theorem is referenced by: ablsimpgfind 19494 |
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