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Mirrors > Home > MPE Home > Th. List > zrhval2 | Structured version Visualization version GIF version |
Description: Alternate value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
zrhval.l | ⊢ 𝐿 = (ℤRHom‘𝑅) |
zrhval2.m | ⊢ · = (.g‘𝑅) |
zrhval2.1 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
zrhval2 | ⊢ (𝑅 ∈ Ring → 𝐿 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zrhval.l | . . 3 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
2 | 1 | zrhval 19904 | . 2 ⊢ 𝐿 = ∪ (ℤring RingHom 𝑅) |
3 | zrhval2.m | . . . . 5 ⊢ · = (.g‘𝑅) | |
4 | eqid 2651 | . . . . 5 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) | |
5 | zrhval2.1 | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
6 | 3, 4, 5 | mulgrhm2 19895 | . . . 4 ⊢ (𝑅 ∈ Ring → (ℤring RingHom 𝑅) = {(𝑛 ∈ ℤ ↦ (𝑛 · 1 ))}) |
7 | 6 | unieqd 4478 | . . 3 ⊢ (𝑅 ∈ Ring → ∪ (ℤring RingHom 𝑅) = ∪ {(𝑛 ∈ ℤ ↦ (𝑛 · 1 ))}) |
8 | zex 11424 | . . . . 5 ⊢ ℤ ∈ V | |
9 | 8 | mptex 6527 | . . . 4 ⊢ (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) ∈ V |
10 | 9 | unisn 4483 | . . 3 ⊢ ∪ {(𝑛 ∈ ℤ ↦ (𝑛 · 1 ))} = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) |
11 | 7, 10 | syl6eq 2701 | . 2 ⊢ (𝑅 ∈ Ring → ∪ (ℤring RingHom 𝑅) = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))) |
12 | 2, 11 | syl5eq 2697 | 1 ⊢ (𝑅 ∈ Ring → 𝐿 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 {csn 4210 ∪ cuni 4468 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 ℤcz 11415 .gcmg 17587 1rcur 18547 Ringcrg 18593 RingHom crh 18760 ℤringzring 19866 ℤRHomczrh 19896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-seq 12842 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-grp 17472 df-minusg 17473 df-mulg 17588 df-subg 17638 df-ghm 17705 df-cmn 18241 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-rnghom 18763 df-subrg 18826 df-cnfld 19795 df-zring 19867 df-zrh 19900 |
This theorem is referenced by: zrhmulg 19906 zrhrhmb 19907 zncyg 19945 zrhchr 30148 zrhre 30191 |
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