![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 19073 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
8 | 7 | 3expa 1115 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ) ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
9 | 8 | an32s 650 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (𝑁 · 𝑋) ∈ 𝑆) |
10 | 9 | 3adantl2 1164 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (𝑁 · 𝑋) ∈ 𝑆) |
11 | oveq1 7420 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 · 𝑋) = (0 · 𝑋)) | |
12 | 5 | 3ad2ant1 1130 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ 𝐵) |
13 | simp3 1135 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
14 | 12, 13 | sseldd 3979 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵) |
15 | mulgnn0subcl.z | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
16 | 1, 15, 2 | mulg0 19061 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 ) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (0 · 𝑋) = 0 ) |
18 | 11, 17 | sylan9eqr 2788 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (𝑁 · 𝑋) = 0 ) |
19 | mulgnn0subcl.c | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑆) | |
20 | 19 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 0 ∈ 𝑆) |
21 | 20 | adantr 479 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → 0 ∈ 𝑆) |
22 | 18, 21 | eqeltrd 2826 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (𝑁 · 𝑋) ∈ 𝑆) |
23 | simp2 1134 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℕ0) | |
24 | elnn0 12517 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
25 | 23, 24 | sylib 217 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
26 | 10, 22, 25 | mpjaodan 956 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ⊆ wss 3946 ‘cfv 6543 (class class class)co 7413 0cc0 11146 ℕcn 12255 ℕ0cn0 12515 Basecbs 17205 +gcplusg 17258 0gc0g 17446 .gcmg 19054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12256 df-n0 12516 df-z 12602 df-uz 12866 df-fz 13530 df-seq 14013 df-mulg 19055 |
This theorem is referenced by: mulgsubcl 19075 mulgnn0cl 19077 submmulgcl 19104 mplbas2 22042 evls1fldgencl 33559 |
Copyright terms: Public domain | W3C validator |