Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrhqima | Structured version Visualization version GIF version |
Description: The ℝHom homomorphism leaves rational numbers unchanged. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
Ref | Expression |
---|---|
rrhqima | ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ((ℝHom‘𝑅)‘𝑄) = ((ℚHom‘𝑅)‘𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
2 | eqid 2818 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
3 | 1, 2 | rrhval 31136 | . . . 4 ⊢ (𝑅 ∈ ℝExt → (ℝHom‘𝑅) = (((topGen‘ran (,))CnExt(TopOpen‘𝑅))‘(ℚHom‘𝑅))) |
4 | 3 | fveq1d 6665 | . . 3 ⊢ (𝑅 ∈ ℝExt → ((ℝHom‘𝑅)‘𝑄) = ((((topGen‘ran (,))CnExt(TopOpen‘𝑅))‘(ℚHom‘𝑅))‘𝑄)) |
5 | 4 | adantr 481 | . 2 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ((ℝHom‘𝑅)‘𝑄) = ((((topGen‘ran (,))CnExt(TopOpen‘𝑅))‘(ℚHom‘𝑅))‘𝑄)) |
6 | uniretop 23298 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
7 | eqid 2818 | . . 3 ⊢ ∪ (TopOpen‘𝑅) = ∪ (TopOpen‘𝑅) | |
8 | retop 23297 | . . . 4 ⊢ (topGen‘ran (,)) ∈ Top | |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → (topGen‘ran (,)) ∈ Top) |
10 | 2 | rrexthaus 31147 | . . . 4 ⊢ (𝑅 ∈ ℝExt → (TopOpen‘𝑅) ∈ Haus) |
11 | 10 | adantr 481 | . . 3 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → (TopOpen‘𝑅) ∈ Haus) |
12 | qssre 12346 | . . . 4 ⊢ ℚ ⊆ ℝ | |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ℚ ⊆ ℝ) |
14 | rrextnrg 31141 | . . . . . . 7 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing) | |
15 | rrextdrg 31142 | . . . . . . 7 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ DivRing) | |
16 | 14, 15 | elind 4168 | . . . . . 6 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ (NrmRing ∩ DivRing)) |
17 | eqid 2818 | . . . . . . 7 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
18 | 17 | rrextnlm 31143 | . . . . . 6 ⊢ (𝑅 ∈ ℝExt → (ℤMod‘𝑅) ∈ NrmMod) |
19 | rrextchr 31144 | . . . . . 6 ⊢ (𝑅 ∈ ℝExt → (chr‘𝑅) = 0) | |
20 | eqid 2818 | . . . . . . 7 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
21 | qqtopn 31151 | . . . . . . 7 ⊢ ((TopOpen‘ℝfld) ↾t ℚ) = (TopOpen‘(ℂfld ↾s ℚ)) | |
22 | 20, 21, 17, 2 | qqhcn 31131 | . . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ (ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (((TopOpen‘ℝfld) ↾t ℚ) Cn (TopOpen‘𝑅))) |
23 | 16, 18, 19, 22 | syl3anc 1363 | . . . . 5 ⊢ (𝑅 ∈ ℝExt → (ℚHom‘𝑅) ∈ (((TopOpen‘ℝfld) ↾t ℚ) Cn (TopOpen‘𝑅))) |
24 | retopn 23909 | . . . . . . . 8 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | |
25 | 24 | eqcomi 2827 | . . . . . . 7 ⊢ (TopOpen‘ℝfld) = (topGen‘ran (,)) |
26 | 25 | oveq1i 7155 | . . . . . 6 ⊢ ((TopOpen‘ℝfld) ↾t ℚ) = ((topGen‘ran (,)) ↾t ℚ) |
27 | 26 | oveq1i 7155 | . . . . 5 ⊢ (((TopOpen‘ℝfld) ↾t ℚ) Cn (TopOpen‘𝑅)) = (((topGen‘ran (,)) ↾t ℚ) Cn (TopOpen‘𝑅)) |
28 | 23, 27 | eleqtrdi 2920 | . . . 4 ⊢ (𝑅 ∈ ℝExt → (ℚHom‘𝑅) ∈ (((topGen‘ran (,)) ↾t ℚ) Cn (TopOpen‘𝑅))) |
29 | 28 | adantr 481 | . . 3 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → (ℚHom‘𝑅) ∈ (((topGen‘ran (,)) ↾t ℚ) Cn (TopOpen‘𝑅))) |
30 | simpr 485 | . . 3 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → 𝑄 ∈ ℚ) | |
31 | 6, 7, 9, 11, 13, 29, 30 | cnextfres 22605 | . 2 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ((((topGen‘ran (,))CnExt(TopOpen‘𝑅))‘(ℚHom‘𝑅))‘𝑄) = ((ℚHom‘𝑅)‘𝑄)) |
32 | 5, 31 | eqtrd 2853 | 1 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ((ℝHom‘𝑅)‘𝑄) = ((ℚHom‘𝑅)‘𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 ∪ cuni 4830 ran crn 5549 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 ℚcq 12336 (,)cioo 12726 ↾s cress 16472 ↾t crest 16682 TopOpenctopn 16683 topGenctg 16699 DivRingcdr 19431 ℂfldccnfld 20473 ℤModczlm 20576 chrcchr 20577 ℝfldcrefld 20676 Topctop 21429 Cn ccn 21760 Hauscha 21844 CnExtccnext 22595 NrmRingcnrg 23116 NrmModcnlm 23117 ℚHomcqqh 31112 ℝHomcrrh 31133 ℝExt crrext 31134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 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 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 df-numer 16063 df-denom 16064 df-gz 16254 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-plusf 17839 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-od 18585 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-rnghom 19396 df-drng 19433 df-subrg 19462 df-abv 19517 df-lmod 19565 df-scaf 19566 df-sra 19873 df-rgmod 19874 df-nzr 19959 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-zring 20546 df-zrh 20579 df-zlm 20580 df-chr 20581 df-refld 20677 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-cnext 22596 df-tmd 22608 df-tgp 22609 df-trg 22695 df-xms 22857 df-ms 22858 df-tms 22859 df-nm 23119 df-ngp 23120 df-nrg 23122 df-nlm 23123 df-qqh 31113 df-rrh 31135 df-rrext 31139 |
This theorem is referenced by: rrh0 31155 |
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