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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 ∈ Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsgrp.m | . . 3 ⊢ 𝑀 = (ℂfld ↾s ℕ) | |
2 | 1 | nnsgrpmgm 47238 | . 2 ⊢ 𝑀 ∈ Mgm |
3 | nncn 12251 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
4 | nncn 12251 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
5 | nncn 12251 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
6 | addass 11226 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
7 | 3, 4, 5, 6 | syl3an 1158 | . . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
8 | 7 | 3expia 1119 | . . . 4 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℕ → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))) |
9 | 8 | ralrimiv 3142 | . . 3 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
10 | 9 | rgen2 3194 | . 2 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) |
11 | nnsscn 12248 | . . . 4 ⊢ ℕ ⊆ ℂ | |
12 | 1 | cnfldsrngbas 47223 | . . . 4 ⊢ (ℕ ⊆ ℂ → ℕ = (Base‘𝑀)) |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ ℕ = (Base‘𝑀) |
14 | nnex 12249 | . . . 4 ⊢ ℕ ∈ V | |
15 | 1 | cnfldsrngadd 47224 | . . . 4 ⊢ (ℕ ∈ V → + = (+g‘𝑀)) |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝑀) |
17 | 13, 16 | issgrp 18680 | . 2 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))) |
18 | 2, 10, 17 | mpbir2an 710 | 1 ⊢ 𝑀 ∈ Smgrp |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3058 Vcvv 3471 ⊆ wss 3947 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 + caddc 11142 ℕcn 12243 Basecbs 17180 ↾s cress 17209 +gcplusg 17233 Mgmcmgm 18598 Smgrpcsgrp 18678 ℂfldccnfld 21279 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-mgm 18600 df-sgrp 18679 df-cnfld 21280 |
This theorem is referenced by: (None) |
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