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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrngmsgrp | Structured version Visualization version GIF version |
Description: R is a (multiplicative) semigroup. (Contributed by AV, 4-Feb-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
2zrngbas.r | ⊢ 𝑅 = (ℂfld ↾s 𝐸) |
2zrngmmgm.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
2zrngmsgrp | ⊢ 𝑀 ∈ Smgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2zrng.e | . . 3 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
2 | 2zrngbas.r | . . 3 ⊢ 𝑅 = (ℂfld ↾s 𝐸) | |
3 | 2zrngmmgm.1 | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
4 | 1, 2, 3 | 2zrngmmgm 46215 | . 2 ⊢ 𝑀 ∈ Mgm |
5 | elrabi 3639 | . . . . 5 ⊢ (𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑎 ∈ ℤ) | |
6 | elrabi 3639 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑦 ∈ ℤ) | |
7 | elrabi 3639 | . . . . 5 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℤ) | |
8 | 5, 6, 7 | 3anim123i 1151 | . . . 4 ⊢ ((𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ∧ 𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ∧ 𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}) → (𝑎 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ)) |
9 | zcn 12503 | . . . . 5 ⊢ (𝑎 ∈ ℤ → 𝑎 ∈ ℂ) | |
10 | zcn 12503 | . . . . 5 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
11 | zcn 12503 | . . . . 5 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℂ) | |
12 | 9, 10, 11 | 3anim123i 1151 | . . . 4 ⊢ ((𝑎 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑎 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑏 ∈ ℂ)) |
13 | mulass 11138 | . . . 4 ⊢ ((𝑎 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝑎 · 𝑦) · 𝑏) = (𝑎 · (𝑦 · 𝑏))) | |
14 | 8, 12, 13 | 3syl 18 | . . 3 ⊢ ((𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ∧ 𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ∧ 𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}) → ((𝑎 · 𝑦) · 𝑏) = (𝑎 · (𝑦 · 𝑏))) |
15 | 14 | rgen3 3199 | . 2 ⊢ ∀𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}∀𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}∀𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ((𝑎 · 𝑦) · 𝑏) = (𝑎 · (𝑦 · 𝑏)) |
16 | 1, 2 | 2zrngbas 46205 | . . . . 5 ⊢ 𝐸 = (Base‘𝑅) |
17 | 3, 16 | mgpbas 19900 | . . . 4 ⊢ 𝐸 = (Base‘𝑀) |
18 | 1, 17 | eqtr3i 2766 | . . 3 ⊢ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} = (Base‘𝑀) |
19 | 1, 2 | 2zrngmul 46214 | . . . 4 ⊢ · = (.r‘𝑅) |
20 | 3, 19 | mgpplusg 19898 | . . 3 ⊢ · = (+g‘𝑀) |
21 | 18, 20 | issgrp 18546 | . 2 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}∀𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}∀𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ((𝑎 · 𝑦) · 𝑏) = (𝑎 · (𝑦 · 𝑏)))) |
22 | 4, 15, 21 | mpbir2an 709 | 1 ⊢ 𝑀 ∈ Smgrp |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 {crab 3407 ‘cfv 6496 (class class class)co 7356 ℂcc 11048 · cmul 11055 2c2 12207 ℤcz 12498 Basecbs 17082 ↾s cress 17111 Mgmcmgm 18494 Smgrpcsgrp 18544 mulGrpcmgp 19894 ℂfldccnfld 20794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-mulf 11130 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-fz 13424 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-starv 17147 df-tset 17151 df-ple 17152 df-ds 17154 df-unif 17155 df-mgm 18496 df-sgrp 18545 df-mgp 19895 df-cnfld 20795 |
This theorem is referenced by: 2zrngALT 46217 |
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