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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 44077 | . 2 ⊢ 𝑀 ∈ Mgm |
3 | nncn 11640 | . . . . . 6 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
4 | nncn 11640 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
5 | nncn 11640 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
6 | addass 10618 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
7 | 3, 4, 5, 6 | syl3an 1156 | . . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
8 | 7 | 3expia 1117 | . . . 4 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ ℕ → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))) |
9 | 8 | ralrimiv 3181 | . . 3 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
10 | 9 | rgen2 3203 | . 2 ⊢ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) |
11 | nnsscn 11637 | . . . 4 ⊢ ℕ ⊆ ℂ | |
12 | 1 | cnfldsrngbas 44030 | . . . 4 ⊢ (ℕ ⊆ ℂ → ℕ = (Base‘𝑀)) |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ ℕ = (Base‘𝑀) |
14 | nnex 11638 | . . . 4 ⊢ ℕ ∈ V | |
15 | 1 | cnfldsrngadd 44031 | . . . 4 ⊢ (ℕ ∈ V → + = (+g‘𝑀)) |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝑀) |
17 | 13, 16 | issgrp 17896 | . 2 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))) |
18 | 2, 10, 17 | mpbir2an 709 | 1 ⊢ 𝑀 ∈ Smgrp |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 + caddc 10534 ℕcn 11632 Basecbs 16477 ↾s cress 16478 +gcplusg 16559 Mgmcmgm 17844 Smgrpcsgrp 17894 ℂfldccnfld 20539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-mgm 17846 df-sgrp 17895 df-cnfld 20540 |
This theorem is referenced by: (None) |
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