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Mirrors > Home > MPE Home > Th. List > gexcl | Structured version Visualization version GIF version |
Description: The exponent of a group is a nonnegative integer. (Contributed by Mario Carneiro, 23-Apr-2016.) |
Ref | Expression |
---|---|
gexcl.1 | ⊢ 𝑋 = (Base‘𝐺) |
gexcl.2 | ⊢ 𝐸 = (gEx‘𝐺) |
Ref | Expression |
---|---|
gexcl | ⊢ (𝐺 ∈ 𝑉 → 𝐸 ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gexcl.1 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
2 | eqid 2772 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | eqid 2772 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | gexcl.2 | . . . . 5 ⊢ 𝐸 = (gEx‘𝐺) | |
5 | eqid 2772 | . . . . 5 ⊢ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦(.g‘𝐺)𝑥) = (0g‘𝐺)} = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦(.g‘𝐺)𝑥) = (0g‘𝐺)} | |
6 | 1, 2, 3, 4, 5 | gexlem1 18477 | . . . 4 ⊢ (𝐺 ∈ 𝑉 → ((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦(.g‘𝐺)𝑥) = (0g‘𝐺)} = ∅) ∨ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦(.g‘𝐺)𝑥) = (0g‘𝐺)})) |
7 | simpl 475 | . . . . 5 ⊢ ((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦(.g‘𝐺)𝑥) = (0g‘𝐺)} = ∅) → 𝐸 = 0) | |
8 | elrabi 3584 | . . . . 5 ⊢ (𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦(.g‘𝐺)𝑥) = (0g‘𝐺)} → 𝐸 ∈ ℕ) | |
9 | 7, 8 | orim12i 892 | . . . 4 ⊢ (((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦(.g‘𝐺)𝑥) = (0g‘𝐺)} = ∅) ∨ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦(.g‘𝐺)𝑥) = (0g‘𝐺)}) → (𝐸 = 0 ∨ 𝐸 ∈ ℕ)) |
10 | 6, 9 | syl 17 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (𝐸 = 0 ∨ 𝐸 ∈ ℕ)) |
11 | 10 | orcomd 857 | . 2 ⊢ (𝐺 ∈ 𝑉 → (𝐸 ∈ ℕ ∨ 𝐸 = 0)) |
12 | elnn0 11707 | . 2 ⊢ (𝐸 ∈ ℕ0 ↔ (𝐸 ∈ ℕ ∨ 𝐸 = 0)) | |
13 | 11, 12 | sylibr 226 | 1 ⊢ (𝐺 ∈ 𝑉 → 𝐸 ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2050 ∀wral 3082 {crab 3086 ∅c0 4172 ‘cfv 6185 (class class class)co 6974 0cc0 10333 ℕcn 11437 ℕ0cn0 11705 Basecbs 16337 0gc0g 16567 .gcmg 18023 gExcgex 18427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-sup 8699 df-inf 8700 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-n0 11706 df-z 11792 df-uz 12057 df-gex 18431 |
This theorem is referenced by: gexod 18484 cyggex2 18783 |
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