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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 769 | . . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝐸 = 1) | |
2 | 1 | oveq1d 7206 | . . . . . . . 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 2736 | . . . . . . . . . 10 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
6 | eqid 2736 | . . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
7 | 3, 4, 5, 6 | gexid 18924 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → (𝐸(.g‘𝐺)𝑥) = (0g‘𝐺)) |
8 | 7 | adantl 485 | . . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → (𝐸(.g‘𝐺)𝑥) = (0g‘𝐺)) |
9 | 3, 5 | mulg1 18453 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → (1(.g‘𝐺)𝑥) = 𝑥) |
10 | 9 | adantl 485 | . . . . . . . 8 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = 𝑥) |
11 | 2, 8, 10 | 3eqtr3rd 2780 | . . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝑥 = (0g‘𝐺)) |
12 | velsn 4543 | . . . . . . 7 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
13 | 11, 12 | sylibr 237 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {(0g‘𝐺)}) |
14 | 13 | ex 416 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → (𝑥 ∈ 𝑋 → 𝑥 ∈ {(0g‘𝐺)})) |
15 | 14 | ssrdv 3893 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 ⊆ {(0g‘𝐺)}) |
16 | 3, 6 | mndidcl 18142 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝑋) |
17 | 16 | adantr 484 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → (0g‘𝐺) ∈ 𝑋) |
18 | 17 | snssd 4708 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → {(0g‘𝐺)} ⊆ 𝑋) |
19 | 15, 18 | eqssd 3904 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 = {(0g‘𝐺)}) |
20 | fvex 6708 | . . . 4 ⊢ (0g‘𝐺) ∈ V | |
21 | 20 | ensn1 8672 | . . 3 ⊢ {(0g‘𝐺)} ≈ 1o |
22 | 19, 21 | eqbrtrdi 5078 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 ≈ 1o) |
23 | simpl 486 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝐺 ∈ Mnd) | |
24 | 1nn 11806 | . . . . 5 ⊢ 1 ∈ ℕ | |
25 | 24 | a1i 11 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 1 ∈ ℕ) |
26 | 9 | adantl 485 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = 𝑥) |
27 | en1eqsn 8882 | . . . . . . . . . 10 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {(0g‘𝐺)}) | |
28 | 16, 27 | sylan 583 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝑋 = {(0g‘𝐺)}) |
29 | 28 | eleq2d 2816 | . . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ {(0g‘𝐺)})) |
30 | 29 | biimpa 480 | . . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {(0g‘𝐺)}) |
31 | 30, 12 | sylib 221 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → 𝑥 = (0g‘𝐺)) |
32 | 26, 31 | eqtrd 2771 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = (0g‘𝐺)) |
33 | 32 | ralrimiva 3095 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → ∀𝑥 ∈ 𝑋 (1(.g‘𝐺)𝑥) = (0g‘𝐺)) |
34 | 3, 4, 5, 6 | gexlem2 18925 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 1 ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 (1(.g‘𝐺)𝑥) = (0g‘𝐺)) → 𝐸 ∈ (1...1)) |
35 | 23, 25, 33, 34 | syl3anc 1373 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝐸 ∈ (1...1)) |
36 | elfz1eq 13088 | . . 3 ⊢ (𝐸 ∈ (1...1) → 𝐸 = 1) | |
37 | 35, 36 | syl 17 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → 𝐸 = 1) |
38 | 22, 37 | impbida 801 | 1 ⊢ (𝐺 ∈ Mnd → (𝐸 = 1 ↔ 𝑋 ≈ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 {csn 4527 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 1oc1o 8173 ≈ cen 8601 1c1 10695 ℕcn 11795 ...cfz 13060 Basecbs 16666 0gc0g 16898 Mndcmnd 18127 .gcmg 18442 gExcgex 18871 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-seq 13540 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mulg 18443 df-gex 18875 |
This theorem is referenced by: pgpfac1lem3a 19417 pgpfaclem3 19424 |
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