MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qsssubdrg Structured version   Visualization version   GIF version
Theorem qsssubdrg 21204
Description: The rational numbers are a subset of any subfield of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
qsssubdrg ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) β†’ β„š βŠ† 𝑅)
Proof of Theorem qsssubdrg
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elq 12938 . . 3 (𝑧 ∈ β„š ↔ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„• 𝑧 = (π‘₯ / 𝑦))
2 drngring 20507 . . . . . . . 8 ((β„‚fld β†Ύs 𝑅) ∈ DivRing β†’ (β„‚fld β†Ύs 𝑅) ∈ Ring)
32ad2antlr 723 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (β„‚fld β†Ύs 𝑅) ∈ Ring)
4 zsssubrg 21203 . . . . . . . . . 10 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ β„€ βŠ† 𝑅)
54ad2antrr 722 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ β„€ βŠ† 𝑅)
6 eqid 2730 . . . . . . . . . . 11 (β„‚fld β†Ύs 𝑅) = (β„‚fld β†Ύs 𝑅)
76subrgbas 20471 . . . . . . . . . 10 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ 𝑅 = (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
87ad2antrr 722 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑅 = (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
95, 8sseqtrd 4021 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ β„€ βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
10 simprl 767 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ π‘₯ ∈ β„€)
119, 10sseldd 3982 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ π‘₯ ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
12 nnz 12583 . . . . . . . . . 10 (𝑦 ∈ β„• β†’ 𝑦 ∈ β„€)
1312ad2antll 725 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 ∈ β„€)
149, 13sseldd 3982 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
15 nnne0 12250 . . . . . . . . . 10 (𝑦 ∈ β„• β†’ 𝑦 β‰  0)
1615ad2antll 725 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 β‰  0)
17 cnfld0 21169 . . . . . . . . . . 11 0 = (0gβ€˜β„‚fld)
186, 17subrg0 20469 . . . . . . . . . 10 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ 0 = (0gβ€˜(β„‚fld β†Ύs 𝑅)))
1918ad2antrr 722 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 0 = (0gβ€˜(β„‚fld β†Ύs 𝑅)))
2016, 19neeqtrd 3008 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 β‰  (0gβ€˜(β„‚fld β†Ύs 𝑅)))
21 eqid 2730 . . . . . . . . . 10 (Baseβ€˜(β„‚fld β†Ύs 𝑅)) = (Baseβ€˜(β„‚fld β†Ύs 𝑅))
22 eqid 2730 . . . . . . . . . 10 (Unitβ€˜(β„‚fld β†Ύs 𝑅)) = (Unitβ€˜(β„‚fld β†Ύs 𝑅))
23 eqid 2730 . . . . . . . . . 10 (0gβ€˜(β„‚fld β†Ύs 𝑅)) = (0gβ€˜(β„‚fld β†Ύs 𝑅))
2421, 22, 23drngunit 20505 . . . . . . . . 9 ((β„‚fld β†Ύs 𝑅) ∈ DivRing β†’ (𝑦 ∈ (Unitβ€˜(β„‚fld β†Ύs 𝑅)) ↔ (𝑦 ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)) ∧ 𝑦 β‰  (0gβ€˜(β„‚fld β†Ύs 𝑅)))))
2524ad2antlr 723 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (𝑦 ∈ (Unitβ€˜(β„‚fld β†Ύs 𝑅)) ↔ (𝑦 ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)) ∧ 𝑦 β‰  (0gβ€˜(β„‚fld β†Ύs 𝑅)))))
2614, 20, 25mpbir2and 709 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 ∈ (Unitβ€˜(β„‚fld β†Ύs 𝑅)))
27 eqid 2730 . . . . . . . 8 (/rβ€˜(β„‚fld β†Ύs 𝑅)) = (/rβ€˜(β„‚fld β†Ύs 𝑅))
2821, 22, 27dvrcl 20295 . . . . . . 7 (((β„‚fld β†Ύs 𝑅) ∈ Ring ∧ π‘₯ ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)) ∧ 𝑦 ∈ (Unitβ€˜(β„‚fld β†Ύs 𝑅))) β†’ (π‘₯(/rβ€˜(β„‚fld β†Ύs 𝑅))𝑦) ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
293, 11, 26, 28syl3anc 1369 . . . . . 6 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (π‘₯(/rβ€˜(β„‚fld β†Ύs 𝑅))𝑦) ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
30 simpll 763 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑅 ∈ (SubRingβ€˜β„‚fld))
315, 10sseldd 3982 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ π‘₯ ∈ 𝑅)
32 cnflddiv 21175 . . . . . . . 8 / = (/rβ€˜β„‚fld)
336, 32, 22, 27subrgdv 20479 . . . . . . 7 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ π‘₯ ∈ 𝑅 ∧ 𝑦 ∈ (Unitβ€˜(β„‚fld β†Ύs 𝑅))) β†’ (π‘₯ / 𝑦) = (π‘₯(/rβ€˜(β„‚fld β†Ύs 𝑅))𝑦))
3430, 31, 26, 33syl3anc 1369 . . . . . 6 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (π‘₯ / 𝑦) = (π‘₯(/rβ€˜(β„‚fld β†Ύs 𝑅))𝑦))
3529, 34, 83eltr4d 2846 . . . . 5 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (π‘₯ / 𝑦) ∈ 𝑅)
36 eleq1 2819 . . . . 5 (𝑧 = (π‘₯ / 𝑦) β†’ (𝑧 ∈ 𝑅 ↔ (π‘₯ / 𝑦) ∈ 𝑅))
3735, 36syl5ibrcom 246 . . . 4 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (𝑧 = (π‘₯ / 𝑦) β†’ 𝑧 ∈ 𝑅))
3837rexlimdvva 3209 . . 3 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) β†’ (βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„• 𝑧 = (π‘₯ / 𝑦) β†’ 𝑧 ∈ 𝑅))
391, 38biimtrid 241 . 2 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) β†’ (𝑧 ∈ β„š β†’ 𝑧 ∈ 𝑅))
4039ssrdv 3987 1 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) β†’ β„š βŠ† 𝑅)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆƒwrex 3068   βŠ† wss 3947  β€˜cfv 6542  (class class class)co 7411  0cc0 11112   / cdiv 11875  β„•cn 12216  β„€cz 12562  β„šcq 12936  Basecbs 17148   β†Ύs cress 17177  0gc0g 17389  Ringcrg 20127  Unitcui 20246  /rcdvr 20291  SubRingcsubrg 20457  DivRingcdr 20500  β„‚fldccnfld 21144
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-fz 13489  df-seq 13971  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-mulg 18987  df-subg 19039  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-cring 20130  df-oppr 20225  df-dvdsr 20248  df-unit 20249  df-invr 20279  df-dvr 20292  df-subrg 20459  df-drng 20502  df-cnfld 21145
This theorem is referenced by:  cphqss  24936  resscdrg  25106
  Copyright terms: Public domain W3C validator