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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 788 | . . . . . . . . 9 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝐸 = 1) | |
2 | 1 | oveq1d 6564 | . . . . . . . 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 2610 | . . . . . . . . . 10 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
6 | eqid 2610 | . . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
7 | 3, 4, 5, 6 | gexid 17819 | . . . . . . . . 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 17371 | . . . . . . . . 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 2653 | . . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝑥 = (0g‘𝐺)) |
12 | velsn 4141 | . . . . . . 7 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
13 | 11, 12 | sylibr 223 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {(0g‘𝐺)}) |
14 | 13 | ex 449 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → (𝑥 ∈ 𝑋 → 𝑥 ∈ {(0g‘𝐺)})) |
15 | 14 | ssrdv 3574 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 ⊆ {(0g‘𝐺)}) |
16 | 3, 6 | mndidcl 17131 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝑋) |
17 | 16 | adantr 480 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → (0g‘𝐺) ∈ 𝑋) |
18 | 17 | snssd 4281 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → {(0g‘𝐺)} ⊆ 𝑋) |
19 | 15, 18 | eqssd 3585 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 = {(0g‘𝐺)}) |
20 | fvex 6113 | . . . 4 ⊢ (0g‘𝐺) ∈ V | |
21 | 20 | ensn1 7906 | . . 3 ⊢ {(0g‘𝐺)} ≈ 1𝑜 |
22 | 19, 21 | syl6eqbr 4622 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 ≈ 1𝑜) |
23 | simpl 472 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1𝑜) → 𝐺 ∈ Mnd) | |
24 | 1nn 10908 | . . . . 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 8075 | . . . . . . . . . 10 ⊢ (((0g‘𝐺) ∈ 𝑋 ∧ 𝑋 ≈ 1𝑜) → 𝑋 = {(0g‘𝐺)}) | |
28 | 16, 27 | sylan 487 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1𝑜) → 𝑋 = {(0g‘𝐺)}) |
29 | 28 | eleq2d 2673 | . . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1𝑜) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ {(0g‘𝐺)})) |
30 | 29 | biimpa 500 | . . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1𝑜) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ {(0g‘𝐺)}) |
31 | 30, 12 | sylib 207 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1𝑜) ∧ 𝑥 ∈ 𝑋) → 𝑥 = (0g‘𝐺)) |
32 | 26, 31 | eqtrd 2644 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1𝑜) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = (0g‘𝐺)) |
33 | 32 | ralrimiva 2949 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1𝑜) → ∀𝑥 ∈ 𝑋 (1(.g‘𝐺)𝑥) = (0g‘𝐺)) |
34 | 3, 4, 5, 6 | gexlem2 17820 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 1 ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 (1(.g‘𝐺)𝑥) = (0g‘𝐺)) → 𝐸 ∈ (1...1)) |
35 | 23, 25, 33, 34 | syl3anc 1318 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1𝑜) → 𝐸 ∈ (1...1)) |
36 | elfz1eq 12223 | . . 3 ⊢ (𝐸 ∈ (1...1) → 𝐸 = 1) | |
37 | 35, 36 | syl 17 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1𝑜) → 𝐸 = 1) |
38 | 22, 37 | impbida 873 | 1 ⊢ (𝐺 ∈ Mnd → (𝐸 = 1 ↔ 𝑋 ≈ 1𝑜)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 ≈ cen 7838 1c1 9816 ℕcn 10897 ...cfz 12197 Basecbs 15695 0gc0g 15923 Mndcmnd 17117 .gcmg 17363 gExcgex 17768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-seq 12664 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mulg 17364 df-gex 17772 |
This theorem is referenced by: pgpfac1lem3a 18298 pgpfaclem3 18305 |
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