Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 765 | . . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝐸 = 1) | |
2 | 1 | oveq1d 7160 | . . . . . . . 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 2818 | . . . . . . . . . 10 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
6 | eqid 2818 | . . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
7 | 3, 4, 5, 6 | gexid 18635 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → (𝐸(.g‘𝐺)𝑥) = (0g‘𝐺)) |
8 | 7 | adantl 482 | . . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → (𝐸(.g‘𝐺)𝑥) = (0g‘𝐺)) |
9 | 3, 5 | mulg1 18173 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → (1(.g‘𝐺)𝑥) = 𝑥) |
10 | 9 | adantl 482 | . . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = 𝑥) |
11 | 2, 8, 10 | 3eqtr3rd 2862 | . . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝑥 = (0g‘𝐺)) |
12 | velsn 4573 | . . . . . . 7 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
13 | 11, 12 | sylibr 235 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {(0g‘𝐺)}) |
14 | 13 | ex 413 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → (𝑥 ∈ 𝑋 → 𝑥 ∈ {(0g‘𝐺)})) |
15 | 14 | ssrdv 3970 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 ⊆ {(0g‘𝐺)}) |
16 | 3, 6 | mndidcl 17914 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝑋) |
17 | 16 | adantr 481 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → (0g‘𝐺) ∈ 𝑋) |
18 | 17 | snssd 4734 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → {(0g‘𝐺)} ⊆ 𝑋) |
19 | 15, 18 | eqssd 3981 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 = {(0g‘𝐺)}) |
20 | fvex 6676 | . . . 4 ⊢ (0g‘𝐺) ∈ V | |
21 | 20 | ensn1 8561 | . . 3 ⊢ {(0g‘𝐺)} ≈ 1o |
22 | 19, 21 | eqbrtrdi 5096 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 ≈ 1o) |
23 | simpl 483 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝐺 ∈ Mnd) | |
24 | 1nn 11637 | . . . . 5 ⊢ 1 ∈ ℕ | |
25 | 24 | a1i 11 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 1 ∈ ℕ) |
26 | 9 | adantl 482 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = 𝑥) |
27 | en1eqsn 8736 | . . . . . . . . . 10 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {(0g‘𝐺)}) | |
28 | 16, 27 | sylan 580 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝑋 = {(0g‘𝐺)}) |
29 | 28 | eleq2d 2895 | . . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ {(0g‘𝐺)})) |
30 | 29 | biimpa 477 | . . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {(0g‘𝐺)}) |
31 | 30, 12 | sylib 219 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → 𝑥 = (0g‘𝐺)) |
32 | 26, 31 | eqtrd 2853 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = (0g‘𝐺)) |
33 | 32 | ralrimiva 3179 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → ∀𝑥 ∈ 𝑋 (1(.g‘𝐺)𝑥) = (0g‘𝐺)) |
34 | 3, 4, 5, 6 | gexlem2 18636 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 1 ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 (1(.g‘𝐺)𝑥) = (0g‘𝐺)) → 𝐸 ∈ (1...1)) |
35 | 23, 25, 33, 34 | syl3anc 1363 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝐸 ∈ (1...1)) |
36 | elfz1eq 12906 | . . 3 ⊢ (𝐸 ∈ (1...1) → 𝐸 = 1) | |
37 | 35, 36 | syl 17 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝐸 = 1) |
38 | 22, 37 | impbida 797 | 1 ⊢ (𝐺 ∈ Mnd → (𝐸 = 1 ↔ 𝑋 ≈ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {csn 4557 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 1oc1o 8084 ≈ cen 8494 1c1 10526 ℕcn 11626 ...cfz 12880 Basecbs 16471 0gc0g 16701 Mndcmnd 17899 .gcmg 18162 gExcgex 18582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-seq 13358 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mulg 18163 df-gex 18586 |
This theorem is referenced by: pgpfac1lem3a 19127 pgpfaclem3 19134 |
Copyright terms: Public domain | W3C validator |