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| 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 7446 | . . . . . . . 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 2737 | . . . . . . . . . 10 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
| 6 | eqid 2737 | . . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 7 | 3, 4, 5, 6 | gexid 19599 | . . . . . . . . 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 19099 | . . . . . . . . 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 2786 | . . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐸 = 1) ∧ 𝑥 ∈ 𝑋) → 𝑥 = (0g‘𝐺)) | 
| 12 | velsn 4642 | . . . . . . 7 ⊢ (𝑥 ∈ {(0g‘𝐺)} ↔ 𝑥 = (0g‘𝐺)) | |
| 13 | 11, 12 | sylibr 234 | . . . . . 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 3989 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 ⊆ {(0g‘𝐺)}) | 
| 16 | 3, 6 | mndidcl 18762 | . . . . . 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 4001 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 = 1) → 𝑋 = {(0g‘𝐺)}) | 
| 20 | fvex 6919 | . . . 4 ⊢ (0g‘𝐺) ∈ V | |
| 21 | 20 | ensn1 9061 | . . 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 12277 | . . . . 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 9308 | . . . . . . . . . 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 2827 | . . . . . . . 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 218 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → 𝑥 = (0g‘𝐺)) | 
| 32 | 26, 31 | eqtrd 2777 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) ∧ 𝑥 ∈ 𝑋) → (1(.g‘𝐺)𝑥) = (0g‘𝐺)) | 
| 33 | 32 | ralrimiva 3146 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ≈ 1o) → ∀𝑥 ∈ 𝑋 (1(.g‘𝐺)𝑥) = (0g‘𝐺)) | 
| 34 | 3, 4, 5, 6 | gexlem2 19600 | . . . 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 13575 | . . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 ≈ cen 8982 1c1 11156 ℕcn 12266 ...cfz 13547 Basecbs 17247 0gc0g 17484 Mndcmnd 18747 .gcmg 19085 gExcgex 19543 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mulg 19086 df-gex 19547 | 
| This theorem is referenced by: pgpfac1lem3a 20096 pgpfaclem3 20103 | 
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