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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 47107 | . 2 ⊢ 𝑀 ∈ Mgm |
3 | nncn 12221 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
4 | nncn 12221 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
5 | nncn 12221 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
6 | addass 11196 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
7 | 3, 4, 5, 6 | syl3an 1157 | . . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
8 | 7 | 3expia 1118 | . . . 4 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℕ → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))) |
9 | 8 | ralrimiv 3139 | . . 3 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
10 | 9 | rgen2 3191 | . 2 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) |
11 | nnsscn 12218 | . . . 4 ⊢ ℕ ⊆ ℂ | |
12 | 1 | cnfldsrngbas 47092 | . . . 4 ⊢ (ℕ ⊆ ℂ → ℕ = (Base‘𝑀)) |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ ℕ = (Base‘𝑀) |
14 | nnex 12219 | . . . 4 ⊢ ℕ ∈ V | |
15 | 1 | cnfldsrngadd 47093 | . . . 4 ⊢ (ℕ ∈ V → + = (+g‘𝑀)) |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝑀) |
17 | 13, 16 | issgrp 18651 | . 2 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))) |
18 | 2, 10, 17 | mpbir2an 708 | 1 ⊢ 𝑀 ∈ Smgrp |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ⊆ wss 3943 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 + caddc 11112 ℕcn 12213 Basecbs 17151 ↾s cress 17180 +gcplusg 17204 Mgmcmgm 18569 Smgrpcsgrp 18649 ℂfldccnfld 21236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-mgm 18571 df-sgrp 18650 df-cnfld 21237 |
This theorem is referenced by: (None) |
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