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Mirrors > Home > MPE Home > Th. List > mulgsubcl | Structured version Visualization version GIF version |
Description: Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
mulgnnsubcl.b | ⊢ 𝐵 = (Base‘𝐺) |
mulgnnsubcl.t | ⊢ · = (.g‘𝐺) |
mulgnnsubcl.p | ⊢ + = (+g‘𝐺) |
mulgnnsubcl.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
mulgnnsubcl.s | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
mulgnnsubcl.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) |
mulgnn0subcl.z | ⊢ 0 = (0g‘𝐺) |
mulgnn0subcl.c | ⊢ (𝜑 → 0 ∈ 𝑆) |
mulgsubcl.i | ⊢ 𝐼 = (invg‘𝐺) |
mulgsubcl.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐼‘𝑥) ∈ 𝑆) |
Ref | Expression |
---|---|
mulgsubcl | ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgnnsubcl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mulgnnsubcl.t | . . . . . 6 ⊢ · = (.g‘𝐺) | |
3 | mulgnnsubcl.p | . . . . . 6 ⊢ + = (+g‘𝐺) | |
4 | mulgnnsubcl.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
5 | mulgnnsubcl.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
6 | mulgnnsubcl.c | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) | |
7 | mulgnn0subcl.z | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
8 | mulgnn0subcl.c | . . . . . 6 ⊢ (𝜑 → 0 ∈ 𝑆) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mulgnn0subcl 18240 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
10 | 9 | 3expa 1114 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0) ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
11 | 10 | an32s 650 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ0) → (𝑁 · 𝑋) ∈ 𝑆) |
12 | 11 | 3adantl2 1163 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ0) → (𝑁 · 𝑋) ∈ 𝑆) |
13 | simp2 1133 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℤ) | |
14 | 13 | adantr 483 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℤ) |
15 | 14 | zcnd 12087 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℂ) |
16 | 15 | negnegd 10987 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → --𝑁 = 𝑁) |
17 | 16 | oveq1d 7170 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (--𝑁 · 𝑋) = (𝑁 · 𝑋)) |
18 | id 22 | . . . . . 6 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ) | |
19 | 5 | 3ad2ant1 1129 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ 𝐵) |
20 | simp3 1134 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
21 | 19, 20 | sseldd 3967 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵) |
22 | mulgsubcl.i | . . . . . . 7 ⊢ 𝐼 = (invg‘𝐺) | |
23 | 1, 2, 22 | mulgnegnn 18237 | . . . . . 6 ⊢ ((-𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (--𝑁 · 𝑋) = (𝐼‘(-𝑁 · 𝑋))) |
24 | 18, 21, 23 | syl2anr 598 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (--𝑁 · 𝑋) = (𝐼‘(-𝑁 · 𝑋))) |
25 | 17, 24 | eqtr3d 2858 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (𝑁 · 𝑋) = (𝐼‘(-𝑁 · 𝑋))) |
26 | fveq2 6669 | . . . . . 6 ⊢ (𝑥 = (-𝑁 · 𝑋) → (𝐼‘𝑥) = (𝐼‘(-𝑁 · 𝑋))) | |
27 | 26 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = (-𝑁 · 𝑋) → ((𝐼‘𝑥) ∈ 𝑆 ↔ (𝐼‘(-𝑁 · 𝑋)) ∈ 𝑆)) |
28 | mulgsubcl.c | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐼‘𝑥) ∈ 𝑆) | |
29 | 28 | ralrimiva 3182 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆) |
30 | 29 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆) |
31 | 30 | adantr 483 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆) |
32 | 1, 2, 3, 4, 5, 6 | mulgnnsubcl 18239 | . . . . . . . 8 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑆) → (-𝑁 · 𝑋) ∈ 𝑆) |
33 | 32 | 3expa 1114 | . . . . . . 7 ⊢ (((𝜑 ∧ -𝑁 ∈ ℕ) ∧ 𝑋 ∈ 𝑆) → (-𝑁 · 𝑋) ∈ 𝑆) |
34 | 33 | an32s 650 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (-𝑁 · 𝑋) ∈ 𝑆) |
35 | 34 | 3adantl2 1163 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (-𝑁 · 𝑋) ∈ 𝑆) |
36 | 27, 31, 35 | rspcdva 3624 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (𝐼‘(-𝑁 · 𝑋)) ∈ 𝑆) |
37 | 25, 36 | eqeltrd 2913 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (𝑁 · 𝑋) ∈ 𝑆) |
38 | 37 | adantrl 714 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝑁 · 𝑋) ∈ 𝑆) |
39 | elznn0nn 11994 | . . 3 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) | |
40 | 13, 39 | sylib 220 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
41 | 12, 38, 40 | mpjaodan 955 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ‘cfv 6354 (class class class)co 7155 ℝcr 10535 -cneg 10870 ℕcn 11637 ℕ0cn0 11896 ℤcz 11980 Basecbs 16482 +gcplusg 16564 0gc0g 16712 invgcminusg 18103 .gcmg 18223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-seq 13369 df-mulg 18224 |
This theorem is referenced by: mulgcl 18244 subgmulgcl 18291 |
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