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Mirrors > Home > MPE Home > Th. List > ablsimpgfindlem2 | Structured version Visualization version GIF version |
Description: Lemma for ablsimpgfind 19232. 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 487 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → (2 · 𝑥) = 0 ) | |
2 | ablsimpgfindlem1.6 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ SimpGrp) | |
3 | 2 | simpggrpd 19217 | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐺 ∈ Grp) |
5 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
6 | 2z 12015 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 2 ∈ ℤ) |
8 | 4, 5, 7 | 3jca 1124 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 2 ∈ ℤ)) |
9 | 8 | adantr 483 | . . . 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 18675 | . . . 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 259 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → (𝑂‘𝑥) ∥ 2) |
17 | 2ne0 11742 | . . . . 5 ⊢ 2 ≠ 0 | |
18 | 17 | a1i 11 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → 2 ≠ 0) |
19 | 18 | neneqd 3021 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → ¬ 2 = 0) |
20 | 0dvds 15630 | . . . 4 ⊢ (2 ∈ ℤ → (0 ∥ 2 ↔ 2 = 0)) | |
21 | 6, 20 | ax-mp 5 | . . 3 ⊢ (0 ∥ 2 ↔ 2 = 0) |
22 | 19, 21 | sylnibr 331 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → ¬ 0 ∥ 2) |
23 | nbrne2 5086 | . 2 ⊢ (((𝑂‘𝑥) ∥ 2 ∧ ¬ 0 ∥ 2) → (𝑂‘𝑥) ≠ 0) | |
24 | 16, 22, 23 | syl2anc 586 | 1 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) = 0 ) → (𝑂‘𝑥) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 0cc0 10537 2c2 11693 ℤcz 11982 ∥ cdvds 15607 Basecbs 16483 0gc0g 16713 Grpcgrp 18103 .gcmg 18224 odcod 18652 Abelcabl 18907 SimpGrpcsimpg 19212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-od 18656 df-simpg 19213 |
This theorem is referenced by: ablsimpgfind 19232 |
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