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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 2737 | . . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,)) | |
| 2 | eqid 2737 | . . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅) | |
| 3 | 1, 2 | rrhval 33997 | . . . 4 ⊢ (𝑅 ∈ ℝExt → (ℝHom‘𝑅) = (((topGen‘ran (,))CnExt(TopOpen‘𝑅))‘(ℚHom‘𝑅))) |
| 4 | 3 | fveq1d 6908 | . . 3 ⊢ (𝑅 ∈ ℝExt → ((ℝHom‘𝑅)‘𝑄) = ((((topGen‘ran (,))CnExt(TopOpen‘𝑅))‘(ℚHom‘𝑅))‘𝑄)) |
| 5 | 4 | adantr 480 | . 2 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ((ℝHom‘𝑅)‘𝑄) = ((((topGen‘ran (,))CnExt(TopOpen‘𝑅))‘(ℚHom‘𝑅))‘𝑄)) |
| 6 | uniretop 24783 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
| 7 | eqid 2737 | . . 3 ⊢ ∪ (TopOpen‘𝑅) = ∪ (TopOpen‘𝑅) | |
| 8 | retop 24782 | . . . 4 ⊢ (topGen‘ran (,)) ∈ Top | |
| 9 | 8 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → (topGen‘ran (,)) ∈ Top) |
| 10 | 2 | rrexthaus 34008 | . . . 4 ⊢ (𝑅 ∈ ℝExt → (TopOpen‘𝑅) ∈ Haus) |
| 11 | 10 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → (TopOpen‘𝑅) ∈ Haus) |
| 12 | qssre 13001 | . . . 4 ⊢ ℚ ⊆ ℝ | |
| 13 | 12 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ℚ ⊆ ℝ) |
| 14 | rrextnrg 34002 | . . . . . . 7 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing) | |
| 15 | rrextdrg 34003 | . . . . . . 7 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ DivRing) | |
| 16 | 14, 15 | elind 4200 | . . . . . 6 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ (NrmRing ∩ DivRing)) |
| 17 | eqid 2737 | . . . . . . 7 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
| 18 | 17 | rrextnlm 34004 | . . . . . 6 ⊢ (𝑅 ∈ ℝExt → (ℤMod‘𝑅) ∈ NrmMod) |
| 19 | rrextchr 34005 | . . . . . 6 ⊢ (𝑅 ∈ ℝExt → (chr‘𝑅) = 0) | |
| 20 | eqid 2737 | . . . . . . 7 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
| 21 | qqtopn 34012 | . . . . . . 7 ⊢ ((TopOpen‘ℝfld) ↾t ℚ) = (TopOpen‘(ℂfld ↾s ℚ)) | |
| 22 | 20, 21, 17, 2 | qqhcn 33992 | . . . . . 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 1373 | . . . . 5 ⊢ (𝑅 ∈ ℝExt → (ℚHom‘𝑅) ∈ (((TopOpen‘ℝfld) ↾t ℚ) Cn (TopOpen‘𝑅))) |
| 24 | retopn 25413 | . . . . . . . 8 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | |
| 25 | 24 | eqcomi 2746 | . . . . . . 7 ⊢ (TopOpen‘ℝfld) = (topGen‘ran (,)) |
| 26 | 25 | oveq1i 7441 | . . . . . 6 ⊢ ((TopOpen‘ℝfld) ↾t ℚ) = ((topGen‘ran (,)) ↾t ℚ) |
| 27 | 26 | oveq1i 7441 | . . . . 5 ⊢ (((TopOpen‘ℝfld) ↾t ℚ) Cn (TopOpen‘𝑅)) = (((topGen‘ran (,)) ↾t ℚ) Cn (TopOpen‘𝑅)) |
| 28 | 23, 27 | eleqtrdi 2851 | . . . 4 ⊢ (𝑅 ∈ ℝExt → (ℚHom‘𝑅) ∈ (((topGen‘ran (,)) ↾t ℚ) Cn (TopOpen‘𝑅))) |
| 29 | 28 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → (ℚHom‘𝑅) ∈ (((topGen‘ran (,)) ↾t ℚ) Cn (TopOpen‘𝑅))) |
| 30 | simpr 484 | . . 3 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → 𝑄 ∈ ℚ) | |
| 31 | 6, 7, 9, 11, 13, 29, 30 | cnextfres 24077 | . 2 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ((((topGen‘ran (,))CnExt(TopOpen‘𝑅))‘(ℚHom‘𝑅))‘𝑄) = ((ℚHom‘𝑅)‘𝑄)) |
| 32 | 5, 31 | eqtrd 2777 | 1 ⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ((ℝHom‘𝑅)‘𝑄) = ((ℚHom‘𝑅)‘𝑄)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 ℚcq 12990 (,)cioo 13387 ↾s cress 17274 ↾t crest 17465 TopOpenctopn 17466 topGenctg 17482 DivRingcdr 20729 ℂfldccnfld 21364 ℤModczlm 21511 chrcchr 21512 ℝfldcrefld 21622 Topctop 22899 Cn ccn 23232 Hauscha 23316 CnExtccnext 24067 NrmRingcnrg 24592 NrmModcnlm 24593 ℚHomcqqh 33971 ℝHomcrrh 33994 ℝExt crrext 33995 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-numer 16772 df-denom 16773 df-gz 16968 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-od 19546 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-drng 20731 df-abv 20810 df-lmod 20860 df-scaf 20861 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-zlm 21515 df-chr 21516 df-refld 21623 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-cnext 24068 df-tmd 24080 df-tgp 24081 df-trg 24168 df-xms 24330 df-ms 24331 df-tms 24332 df-nm 24595 df-ngp 24596 df-nrg 24598 df-nlm 24599 df-qqh 33972 df-rrh 33996 df-rrext 34000 |
| This theorem is referenced by: rrh0 34016 |
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