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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnsgrp | Structured version Visualization version GIF version |
Description: The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.) |
Ref | Expression |
---|---|
nnsgrp.m | ⊢ 𝑀 = (ℂfld ↾s ℕ) |
Ref | Expression |
---|---|
nnsgrp | ⊢ 𝑀 ∈ SGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsgrp.m | . . 3 ⊢ 𝑀 = (ℂfld ↾s ℕ) | |
2 | 1 | nnsgrpmgm 42341 | . 2 ⊢ 𝑀 ∈ Mgm |
3 | nncn 11234 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
4 | nncn 11234 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
5 | nncn 11234 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
6 | addass 10229 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
7 | 3, 4, 5, 6 | syl3an 1163 | . . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
8 | 7 | 3expia 1114 | . . . 4 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℕ → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))) |
9 | 8 | ralrimiv 3114 | . . 3 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
10 | 9 | rgen2a 3126 | . 2 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) |
11 | nnsscn 11231 | . . . 4 ⊢ ℕ ⊆ ℂ | |
12 | 1 | cnfldsrngbas 42294 | . . . 4 ⊢ (ℕ ⊆ ℂ → ℕ = (Base‘𝑀)) |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ ℕ = (Base‘𝑀) |
14 | nnex 11232 | . . . 4 ⊢ ℕ ∈ V | |
15 | 1 | cnfldsrngadd 42295 | . . . 4 ⊢ (ℕ ∈ V → + = (+g‘𝑀)) |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝑀) |
17 | 13, 16 | issgrp 17493 | . 2 ⊢ (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))) |
18 | 2, 10, 17 | mpbir2an 690 | 1 ⊢ 𝑀 ∈ SGrp |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 Vcvv 3351 ⊆ wss 3723 ‘cfv 6030 (class class class)co 6796 ℂcc 10140 + caddc 10145 ℕcn 11226 Basecbs 16064 ↾s cress 16065 +gcplusg 16149 Mgmcmgm 17448 SGrpcsgrp 17491 ℂfldccnfld 19961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-addf 10221 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-7 11290 df-8 11291 df-9 11292 df-n0 11500 df-z 11585 df-dec 11701 df-uz 11894 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-mgm 17450 df-sgrp 17492 df-cnfld 19962 |
This theorem is referenced by: (None) |
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