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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zrhre | Structured version Visualization version GIF version |
Description: The ℤRHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.) |
Ref | Expression |
---|---|
zrhre | ⊢ (ℤRHom‘ℝfld) = ( I ↾ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubdrg 20355 | . . . 4 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
2 | 1 | simpri 481 | . . 3 ⊢ ℝfld ∈ DivRing |
3 | drngring 19150 | . . 3 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
4 | eqid 2778 | . . . 4 ⊢ (ℤRHom‘ℝfld) = (ℤRHom‘ℝfld) | |
5 | eqid 2778 | . . . 4 ⊢ (.g‘ℝfld) = (.g‘ℝfld) | |
6 | re1r 20360 | . . . 4 ⊢ 1 = (1r‘ℝfld) | |
7 | 4, 5, 6 | zrhval2 20257 | . . 3 ⊢ (ℝfld ∈ Ring → (ℤRHom‘ℝfld) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘ℝfld)1))) |
8 | 2, 3, 7 | mp2b 10 | . 2 ⊢ (ℤRHom‘ℝfld) = (𝑛 ∈ ℤ ↦ (𝑛(.g‘ℝfld)1)) |
9 | 1re 10378 | . . . . 5 ⊢ 1 ∈ ℝ | |
10 | remulg 20354 | . . . . 5 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℝ) → (𝑛(.g‘ℝfld)1) = (𝑛 · 1)) | |
11 | 9, 10 | mpan2 681 | . . . 4 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℝfld)1) = (𝑛 · 1)) |
12 | zre 11736 | . . . . 5 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℝ) | |
13 | ax-1rid 10344 | . . . . 5 ⊢ (𝑛 ∈ ℝ → (𝑛 · 1) = 𝑛) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝑛 ∈ ℤ → (𝑛 · 1) = 𝑛) |
15 | 11, 14 | eqtrd 2814 | . . 3 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℝfld)1) = 𝑛) |
16 | 15 | mpteq2ia 4977 | . 2 ⊢ (𝑛 ∈ ℤ ↦ (𝑛(.g‘ℝfld)1)) = (𝑛 ∈ ℤ ↦ 𝑛) |
17 | mptresid 5714 | . 2 ⊢ (𝑛 ∈ ℤ ↦ 𝑛) = ( I ↾ ℤ) | |
18 | 8, 16, 17 | 3eqtri 2806 | 1 ⊢ (ℤRHom‘ℝfld) = ( I ↾ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 ↦ cmpt 4967 I cid 5262 ↾ cres 5359 ‘cfv 6137 (class class class)co 6924 ℝcr 10273 1c1 10275 · cmul 10279 ℤcz 11732 .gcmg 17931 Ringcrg 18938 DivRingcdr 19143 SubRingcsubrg 19172 ℂfldccnfld 20146 ℤRHomczrh 20248 ℝfldcrefld 20351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-fz 12648 df-seq 13124 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-0g 16492 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-mhm 17725 df-grp 17816 df-minusg 17817 df-mulg 17932 df-subg 17979 df-ghm 18046 df-cmn 18585 df-mgp 18881 df-ur 18893 df-ring 18940 df-cring 18941 df-oppr 19014 df-dvdsr 19032 df-unit 19033 df-invr 19063 df-dvr 19074 df-rnghom 19108 df-drng 19145 df-subrg 19174 df-cnfld 20147 df-zring 20219 df-zrh 20252 df-refld 20352 |
This theorem is referenced by: qqhre 30666 |
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