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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrngasgrp | Structured version Visualization version GIF version |
Description: R is an (additive) 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) |
Ref | Expression |
---|---|
2zrngasgrp | ⊢ 𝑅 ∈ 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 | 1, 2 | 2zrngamgm 44217 | . 2 ⊢ 𝑅 ∈ Mgm |
4 | elrabi 3677 | . . . . 5 ⊢ (𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑎 ∈ ℤ) | |
5 | elrabi 3677 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑦 ∈ ℤ) | |
6 | elrabi 3677 | . . . . 5 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℤ) | |
7 | 4, 5, 6 | 3anim123i 1147 | . . . 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)) |
8 | zcn 11989 | . . . . 5 ⊢ (𝑎 ∈ ℤ → 𝑎 ∈ ℂ) | |
9 | zcn 11989 | . . . . 5 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
10 | zcn 11989 | . . . . 5 ⊢ (𝑏 ∈ ℤ → 𝑏 ∈ ℂ) | |
11 | 8, 9, 10 | 3anim123i 1147 | . . . 4 ⊢ ((𝑎 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑎 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑏 ∈ ℂ)) |
12 | addass 10626 | . . . 4 ⊢ ((𝑎 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝑎 + 𝑦) + 𝑏) = (𝑎 + (𝑦 + 𝑏))) | |
13 | 7, 11, 12 | 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))) |
14 | 13 | rgen3 3206 | . 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)) |
15 | 1, 2 | 2zrngbas 44214 | . . . 4 ⊢ 𝐸 = (Base‘𝑅) |
16 | 1, 15 | eqtr3i 2848 | . . 3 ⊢ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} = (Base‘𝑅) |
17 | 1, 2 | 2zrngadd 44215 | . . 3 ⊢ + = (+g‘𝑅) |
18 | 16, 17 | issgrp 17904 | . 2 ⊢ (𝑅 ∈ Smgrp ↔ (𝑅 ∈ 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)))) |
19 | 3, 14, 18 | mpbir2an 709 | 1 ⊢ 𝑅 ∈ Smgrp |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 {crab 3144 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 + caddc 10542 · cmul 10544 2c2 11695 ℤcz 11984 Basecbs 16485 ↾s cress 16486 Mgmcmgm 17852 Smgrpcsgrp 17902 ℂfldccnfld 20547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-mgm 17854 df-sgrp 17903 df-cnfld 20548 |
This theorem is referenced by: 2zrngamnd 44219 |
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