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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrngmul | Structured version Visualization version GIF version |
Description: The ring multiplication operation of R is the multiplication on complex numbers. (Contributed by AV, 31-Jan-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
2zrngbas.r | ⊢ 𝑅 = (ℂfld ↾s 𝐸) |
Ref | Expression |
---|---|
2zrngmul | ⊢ · = (.r‘𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2zrng.e | . . 3 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
2 | zex 12182 | . . 3 ⊢ ℤ ∈ V | |
3 | 1, 2 | rabex2 5224 | . 2 ⊢ 𝐸 ∈ V |
4 | 2zrngbas.r | . . 3 ⊢ 𝑅 = (ℂfld ↾s 𝐸) | |
5 | 4 | cnfldsrngmul 44996 | . 2 ⊢ (𝐸 ∈ V → · = (.r‘𝑅)) |
6 | 3, 5 | ax-mp 5 | 1 ⊢ · = (.r‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ∃wrex 3059 {crab 3062 Vcvv 3405 ‘cfv 6377 (class class class)co 7210 · cmul 10731 2c2 11882 ℤcz 12173 ↾s cress 16781 .rcmulr 16800 ℂfldccnfld 20360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-mulf 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 df-n0 12088 df-z 12174 df-dec 12291 df-uz 12436 df-fz 13093 df-struct 16697 df-sets 16714 df-slot 16732 df-ndx 16742 df-base 16758 df-ress 16782 df-plusg 16812 df-mulr 16813 df-starv 16814 df-tset 16818 df-ple 16819 df-ds 16821 df-unif 16822 df-cnfld 20361 |
This theorem is referenced by: 2zrngmmgm 45175 2zrngmsgrp 45176 2zrngALT 45177 2zrngnring 45181 |
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