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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 19104 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
| 8 | 7 | 3expa 1119 | . . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ) ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
| 9 | 8 | an32s 652 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (𝑁 · 𝑋) ∈ 𝑆) |
| 10 | 9 | 3adantl2 1168 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (𝑁 · 𝑋) ∈ 𝑆) |
| 11 | oveq1 7438 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 · 𝑋) = (0 · 𝑋)) | |
| 12 | 5 | 3ad2ant1 1134 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ 𝐵) |
| 13 | simp3 1139 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
| 14 | 12, 13 | sseldd 3984 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵) |
| 15 | mulgnn0subcl.z | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
| 16 | 1, 15, 2 | mulg0 19092 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 ) |
| 17 | 14, 16 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (0 · 𝑋) = 0 ) |
| 18 | 11, 17 | sylan9eqr 2799 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (𝑁 · 𝑋) = 0 ) |
| 19 | mulgnn0subcl.c | . . . . 5 ⊢ (𝜑 → 0 ∈ 𝑆) | |
| 20 | 19 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 0 ∈ 𝑆) |
| 21 | 20 | adantr 480 | . . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → 0 ∈ 𝑆) |
| 22 | 18, 21 | eqeltrd 2841 | . 2 ⊢ (((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (𝑁 · 𝑋) ∈ 𝑆) |
| 23 | simp2 1138 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℕ0) | |
| 24 | elnn0 12528 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
| 25 | 23, 24 | sylib 218 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
| 26 | 10, 22, 25 | mpjaodan 961 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 +gcplusg 17297 0gc0g 17484 .gcmg 19085 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 df-mulg 19086 |
| This theorem is referenced by: mulgsubcl 19106 mulgnn0cl 19108 submmulgcl 19135 mplbas2 22060 evls1fldgencl 33720 |
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