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Mirrors > Home > MPE Home > Th. List > gex1 | Structured version Visualization version GIF version |
Description: A group or monoid has exponent 1 iff it is trivial. (Contributed by Mario Carneiro, 24-Apr-2016.) |
Ref | Expression |
---|---|
gexcl2.1 | ⊢ 𝑋 = (Base‘𝐺) |
gexcl2.2 | ⊢ 𝐸 = (gEx‘𝐺) |
Ref | Expression |
---|---|
gex1 | ⊢ (𝐺 ∈ Mnd → (𝐸 = 1 ↔ 𝑋 ≈ 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 768 | . . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝐸 = 1) | |
2 | 1 | oveq1d 7430 | . . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → (𝐸(.g‘𝐺)𝑥) = (1(.g‘𝐺)𝑥)) |
3 | gexcl2.1 | . . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝐺) | |
4 | gexcl2.2 | . . . . . . . . . 10 ⊢ 𝐸 = (gEx‘𝐺) | |
5 | eqid 2728 | . . . . . . . . . 10 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
6 | eqid 2728 | . . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
7 | 3, 4, 5, 6 | gexid 19530 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → (𝐸(.g‘𝐺)𝑥) = (0g‘𝐺)) |
8 | 7 | adantl 481 | . . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → (𝐸(.g‘𝐺)𝑥) = (0g‘𝐺)) |
9 | 3, 5 | mulg1 19030 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → (1(.g‘𝐺)𝑥) = 𝑥) |
10 | 9 | adantl 481 | . . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = 𝑥) |
11 | 2, 8, 10 | 3eqtr3rd 2777 | . . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝑥 = (0g‘𝐺)) |
12 | velsn 4641 | . . . . . . 7 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
13 | 11, 12 | sylibr 233 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {(0g‘𝐺)}) |
14 | 13 | ex 412 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → (𝑥 ∈ 𝑋 → 𝑥 ∈ {(0g‘𝐺)})) |
15 | 14 | ssrdv 3985 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 ⊆ {(0g‘𝐺)}) |
16 | 3, 6 | mndidcl 18703 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝑋) |
17 | 16 | adantr 480 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → (0g‘𝐺) ∈ 𝑋) |
18 | 17 | snssd 4809 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → {(0g‘𝐺)} ⊆ 𝑋) |
19 | 15, 18 | eqssd 3996 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 = {(0g‘𝐺)}) |
20 | fvex 6905 | . . . 4 ⊢ (0g‘𝐺) ∈ V | |
21 | 20 | ensn1 9036 | . . 3 ⊢ {(0g‘𝐺)} ≈ 1o |
22 | 19, 21 | eqbrtrdi 5182 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 ≈ 1o) |
23 | simpl 482 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝐺 ∈ Mnd) | |
24 | 1nn 12248 | . . . . 5 ⊢ 1 ∈ ℕ | |
25 | 24 | a1i 11 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 1 ∈ ℕ) |
26 | 9 | adantl 481 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = 𝑥) |
27 | en1eqsn 9293 | . . . . . . . . . 10 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {(0g‘𝐺)}) | |
28 | 16, 27 | sylan 579 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝑋 = {(0g‘𝐺)}) |
29 | 28 | eleq2d 2815 | . . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ {(0g‘𝐺)})) |
30 | 29 | biimpa 476 | . . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {(0g‘𝐺)}) |
31 | 30, 12 | sylib 217 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → 𝑥 = (0g‘𝐺)) |
32 | 26, 31 | eqtrd 2768 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = (0g‘𝐺)) |
33 | 32 | ralrimiva 3142 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → ∀𝑥 ∈ 𝑋 (1(.g‘𝐺)𝑥) = (0g‘𝐺)) |
34 | 3, 4, 5, 6 | gexlem2 19531 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 1 ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 (1(.g‘𝐺)𝑥) = (0g‘𝐺)) → 𝐸 ∈ (1...1)) |
35 | 23, 25, 33, 34 | syl3anc 1369 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝐸 ∈ (1...1)) |
36 | elfz1eq 13539 | . . 3 ⊢ (𝐸 ∈ (1...1) → 𝐸 = 1) | |
37 | 35, 36 | syl 17 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝐸 = 1) |
38 | 22, 37 | impbida 800 | 1 ⊢ (𝐺 ∈ Mnd → (𝐸 = 1 ↔ 𝑋 ≈ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 {csn 4625 class class class wbr 5143 ‘cfv 6543 (class class class)co 7415 1oc1o 8474 ≈ cen 8955 1c1 11134 ℕcn 12237 ...cfz 13511 Basecbs 17174 0gc0g 17415 Mndcmnd 18688 .gcmg 19017 gExcgex 19474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 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 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-sup 9460 df-inf 9461 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-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-seq 13994 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mulg 19018 df-gex 19478 |
This theorem is referenced by: pgpfac1lem3a 20027 pgpfaclem3 20034 |
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