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Mirrors > Home > MPE Home > Th. List > mulgnn0subcl | Structured version Visualization version GIF version |
Description: Closure of the group multiple (exponentiation) operation in a submonoid. (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) |
Ref | Expression |
---|---|
mulgnn0subcl | ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 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 | 1, 2, 3, 4, 5, 6 | mulgnnsubcl 17553 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
8 | 7 | 3expa 1265 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ) ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
9 | 8 | an32s 846 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (𝑁 · 𝑋) ∈ 𝑆) |
10 | 9 | 3adantl2 1218 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (𝑁 · 𝑋) ∈ 𝑆) |
11 | oveq1 6657 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 · 𝑋) = (0 · 𝑋)) | |
12 | 5 | 3ad2ant1 1082 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ 𝐵) |
13 | simp3 1063 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
14 | 12, 13 | sseldd 3604 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵) |
15 | mulgnn0subcl.z | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
16 | 1, 15, 2 | mulg0 17546 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 ) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (0 · 𝑋) = 0 ) |
18 | 11, 17 | sylan9eqr 2678 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (𝑁 · 𝑋) = 0 ) |
19 | mulgnn0subcl.c | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑆) | |
20 | 19 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 0 ∈ 𝑆) |
21 | 20 | adantr 481 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → 0 ∈ 𝑆) |
22 | 18, 21 | eqeltrd 2701 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (𝑁 · 𝑋) ∈ 𝑆) |
23 | simp2 1062 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℕ0) | |
24 | elnn0 11294 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
25 | 23, 24 | sylib 208 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
26 | 10, 22, 25 | mpjaodan 827 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ℕcn 11020 ℕ0cn0 11292 Basecbs 15857 +gcplusg 15941 0gc0g 16100 .gcmg 17540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 df-mulg 17541 |
This theorem is referenced by: mulgsubcl 17555 mulgnn0cl 17558 submmulgcl 17585 mplbas2 19470 |
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