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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 18505 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
10 | 9 | 3expa 1120 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0) ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
11 | 10 | an32s 652 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ0) → (𝑁 · 𝑋) ∈ 𝑆) |
12 | 11 | 3adantl2 1169 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ0) → (𝑁 · 𝑋) ∈ 𝑆) |
13 | simp2 1139 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℤ) | |
14 | 13 | adantr 484 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℤ) |
15 | 14 | zcnd 12283 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℂ) |
16 | 15 | negnegd 11180 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → --𝑁 = 𝑁) |
17 | 16 | oveq1d 7228 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (--𝑁 · 𝑋) = (𝑁 · 𝑋)) |
18 | id 22 | . . . . . 6 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ) | |
19 | 5 | 3ad2ant1 1135 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ 𝐵) |
20 | simp3 1140 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
21 | 19, 20 | sseldd 3902 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵) |
22 | mulgsubcl.i | . . . . . . 7 ⊢ 𝐼 = (invg‘𝐺) | |
23 | 1, 2, 22 | mulgnegnn 18502 | . . . . . 6 ⊢ ((-𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (--𝑁 · 𝑋) = (𝐼‘(-𝑁 · 𝑋))) |
24 | 18, 21, 23 | syl2anr 600 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (--𝑁 · 𝑋) = (𝐼‘(-𝑁 · 𝑋))) |
25 | 17, 24 | eqtr3d 2779 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (𝑁 · 𝑋) = (𝐼‘(-𝑁 · 𝑋))) |
26 | fveq2 6717 | . . . . . 6 ⊢ (𝑥 = (-𝑁 · 𝑋) → (𝐼‘𝑥) = (𝐼‘(-𝑁 · 𝑋))) | |
27 | 26 | eleq1d 2822 | . . . . 5 ⊢ (𝑥 = (-𝑁 · 𝑋) → ((𝐼‘𝑥) ∈ 𝑆 ↔ (𝐼‘(-𝑁 · 𝑋)) ∈ 𝑆)) |
28 | mulgsubcl.c | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐼‘𝑥) ∈ 𝑆) | |
29 | 28 | ralrimiva 3105 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆) |
30 | 29 | 3ad2ant1 1135 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆) |
31 | 30 | adantr 484 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → ∀𝑥 ∈ 𝑆 (𝐼‘𝑥) ∈ 𝑆) |
32 | 1, 2, 3, 4, 5, 6 | mulgnnsubcl 18504 | . . . . . . . 8 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑆) → (-𝑁 · 𝑋) ∈ 𝑆) |
33 | 32 | 3expa 1120 | . . . . . . 7 ⊢ (((𝜑 ∧ -𝑁 ∈ ℕ) ∧ 𝑋 ∈ 𝑆) → (-𝑁 · 𝑋) ∈ 𝑆) |
34 | 33 | an32s 652 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (-𝑁 · 𝑋) ∈ 𝑆) |
35 | 34 | 3adantl2 1169 | . . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (-𝑁 · 𝑋) ∈ 𝑆) |
36 | 27, 31, 35 | rspcdva 3539 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (𝐼‘(-𝑁 · 𝑋)) ∈ 𝑆) |
37 | 25, 36 | eqeltrd 2838 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ -𝑁 ∈ ℕ) → (𝑁 · 𝑋) ∈ 𝑆) |
38 | 37 | adantrl 716 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) → (𝑁 · 𝑋) ∈ 𝑆) |
39 | elznn0nn 12190 | . . 3 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) | |
40 | 13, 39 | sylib 221 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
41 | 12, 38, 40 | mpjaodan 959 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ⊆ wss 3866 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 -cneg 11063 ℕcn 11830 ℕ0cn0 12090 ℤcz 12176 Basecbs 16760 +gcplusg 16802 0gc0g 16944 invgcminusg 18366 .gcmg 18488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-seq 13575 df-mulg 18489 |
This theorem is referenced by: mulgcl 18509 subgmulgcl 18556 |
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