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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 18039 | . 2 ⊢ (𝐺 ∈ 𝑉 → 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
7 | eqeq2 2662 | . . . 4 ⊢ (0 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) → (𝐸 = 0 ↔ 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))) | |
8 | 7 | imbi1d 330 | . . 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 2662 | . . . 4 ⊢ (inf(𝐼, ℝ, < ) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) → (𝐸 = inf(𝐼, ℝ, < ) ↔ 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))) | |
10 | 9 | imbi1d 330 | . . 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 399 | . . . . 5 ⊢ ((𝐸 = 0 ∧ 𝐼 = ∅) → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) | |
12 | 11 | expcom 450 | . . . 4 ⊢ (𝐼 = ∅ → (𝐸 = 0 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼))) |
13 | 12 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐼 = ∅) → (𝐸 = 0 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼))) |
14 | ssrab2 3720 | . . . . . . 7 ⊢ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ⊆ ℕ | |
15 | nnuz 11761 | . . . . . . . 8 ⊢ ℕ = (ℤ≥‘1) | |
16 | 15 | eqcomi 2660 | . . . . . . 7 ⊢ (ℤ≥‘1) = ℕ |
17 | 14, 5, 16 | 3sstr4i 3677 | . . . . . 6 ⊢ 𝐼 ⊆ (ℤ≥‘1) |
18 | df-ne 2824 | . . . . . . . 8 ⊢ (𝐼 ≠ ∅ ↔ ¬ 𝐼 = ∅) | |
19 | 18 | biimpri 218 | . . . . . . 7 ⊢ (¬ 𝐼 = ∅ → 𝐼 ≠ ∅) |
20 | 19 | adantl 481 | . . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅) → 𝐼 ≠ ∅) |
21 | infssuzcl 11810 | . . . . . 6 ⊢ ((𝐼 ⊆ (ℤ≥‘1) ∧ 𝐼 ≠ ∅) → inf(𝐼, ℝ, < ) ∈ 𝐼) | |
22 | 17, 20, 21 | sylancr 696 | . . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅) → inf(𝐼, ℝ, < ) ∈ 𝐼) |
23 | eleq1a 2725 | . . . . 5 ⊢ (inf(𝐼, ℝ, < ) ∈ 𝐼 → (𝐸 = inf(𝐼, ℝ, < ) → 𝐸 ∈ 𝐼)) | |
24 | 22, 23 | syl 17 | . . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅) → (𝐸 = inf(𝐼, ℝ, < ) → 𝐸 ∈ 𝐼)) |
25 | olc 398 | . . . 4 ⊢ (𝐸 ∈ 𝐼 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) | |
26 | 24, 25 | syl6 35 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ ¬ 𝐼 = ∅) → (𝐸 = inf(𝐼, ℝ, < ) → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼))) |
27 | 8, 10, 13, 26 | ifbothda 4156 | . 2 ⊢ (𝐺 ∈ 𝑉 → (𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼))) |
28 | 6, 27 | mpd 15 | 1 ⊢ (𝐺 ∈ 𝑉 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∀wral 2941 {crab 2945 ⊆ wss 3607 ∅c0 3948 ifcif 4119 ‘cfv 5926 (class class class)co 6690 infcinf 8388 ℝcr 9973 0cc0 9974 1c1 9975 < clt 10112 ℕcn 11058 ℤ≥cuz 11725 Basecbs 15904 0gc0g 16147 .gcmg 17587 gExcgex 17991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-gex 17995 |
This theorem is referenced by: gexcl 18041 gexid 18042 gexdvds 18045 |
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