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Mirrors > Home > MPE Home > Th. List > gexlem1 | Structured version Visualization version GIF version |
Description: The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 23-Apr-2016.) (Proof shortened by AV, 26-Sep-2020.) |
Ref | Expression |
---|---|
gexval.1 | ⊢ 𝑋 = (Base‘𝐺) |
gexval.2 | ⊢ · = (.g‘𝐺) |
gexval.3 | ⊢ 0 = (0g‘𝐺) |
gexval.4 | ⊢ 𝐸 = (gEx‘𝐺) |
gexval.i | ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } |
Ref | Expression |
---|---|
gexlem1 | ⊢ (𝐺 ∈ 𝑉 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gexval.1 | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
2 | gexval.2 | . . 3 ⊢ · = (.g‘𝐺) | |
3 | gexval.3 | . . 3 ⊢ 0 = (0g‘𝐺) | |
4 | gexval.4 | . . 3 ⊢ 𝐸 = (gEx‘𝐺) | |
5 | gexval.i | . . 3 ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } | |
6 | 1, 2, 3, 4, 5 | gexval 18476 | . 2 ⊢ (𝐺 ∈ 𝑉 → 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
7 | eqeq2 2791 | . . . 4 ⊢ (0 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) → (𝐸 = 0 ↔ 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))) | |
8 | 7 | imbi1d 334 | . . 3 ⊢ (0 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) → ((𝐸 = 0 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) ↔ (𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)))) |
9 | eqeq2 2791 | . . . 4 ⊢ (inf(𝐼, ℝ, < ) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) → (𝐸 = inf(𝐼, ℝ, < ) ↔ 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))) | |
10 | 9 | imbi1d 334 | . . 3 ⊢ (inf(𝐼, ℝ, < ) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) → ((𝐸 = inf(𝐼, ℝ, < ) → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) ↔ (𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)))) |
11 | orc 854 | . . . . 5 ⊢ ((𝐸 = 0 ∧ 𝐼 = ∅) → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) | |
12 | 11 | expcom 406 | . . . 4 ⊢ (𝐼 = ∅ → (𝐸 = 0 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼))) |
13 | 12 | adantl 474 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐼 = ∅) → (𝐸 = 0 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼))) |
14 | ssrab2 3948 | . . . . . . 7 ⊢ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ⊆ ℕ | |
15 | nnuz 12101 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
16 | 15 | eqcomi 2789 | . . . . . . 7 ⊢ (ℤ≥‘1) = ℕ |
17 | 14, 5, 16 | 3sstr4i 3902 | . . . . . 6 ⊢ 𝐼 ⊆ (ℤ≥‘1) |
18 | neqne 2977 | . . . . . . 7 ⊢ (¬ 𝐼 = ∅ → 𝐼 ≠ ∅) | |
19 | 18 | adantl 474 | . . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅) → 𝐼 ≠ ∅) |
20 | infssuzcl 12152 | . . . . . 6 ⊢ ((𝐼 ⊆ (ℤ≥‘1) ∧ 𝐼 ≠ ∅) → inf(𝐼, ℝ, < ) ∈ 𝐼) | |
21 | 17, 19, 20 | sylancr 579 | . . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅) → inf(𝐼, ℝ, < ) ∈ 𝐼) |
22 | eleq1a 2863 | . . . . 5 ⊢ (inf(𝐼, ℝ, < ) ∈ 𝐼 → (𝐸 = inf(𝐼, ℝ, < ) → 𝐸 ∈ 𝐼)) | |
23 | 21, 22 | syl 17 | . . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅) → (𝐸 = inf(𝐼, ℝ, < ) → 𝐸 ∈ 𝐼)) |
24 | olc 855 | . . . 4 ⊢ (𝐸 ∈ 𝐼 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) | |
25 | 23, 24 | syl6 35 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅) → (𝐸 = inf(𝐼, ℝ, < ) → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼))) |
26 | 8, 10, 13, 25 | ifbothda 4390 | . 2 ⊢ (𝐺 ∈ 𝑉 → (𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼))) |
27 | 6, 26 | mpd 15 | 1 ⊢ (𝐺 ∈ 𝑉 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∨ wo 834 = wceq 1508 ∈ wcel 2051 ≠ wne 2969 ∀wral 3090 {crab 3094 ⊆ wss 3831 ∅c0 4181 ifcif 4353 ‘cfv 6193 (class class class)co 6982 infcinf 8706 ℝcr 10340 0cc0 10341 1c1 10342 < clt 10480 ℕcn 11445 ℤ≥cuz 12064 Basecbs 16345 0gc0g 16575 .gcmg 18023 gExcgex 18427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-sup 8707 df-inf 8708 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-nn 11446 df-n0 11714 df-z 11800 df-uz 12065 df-gex 18431 |
This theorem is referenced by: gexcl 18478 gexid 18479 gexdvds 18482 |
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