MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qsssubdrg Structured version   Visualization version   GIF version

Theorem qsssubdrg 20763
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 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)

Proof of Theorem qsssubdrg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elq 12791 . . 3 (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
2 drngring 20100 . . . . . . . 8 ((ℂflds 𝑅) ∈ DivRing → (ℂflds 𝑅) ∈ Ring)
32ad2antlr 724 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (ℂflds 𝑅) ∈ Ring)
4 zsssubrg 20762 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
54ad2antrr 723 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ 𝑅)
6 eqid 2736 . . . . . . . . . . 11 (ℂflds 𝑅) = (ℂflds 𝑅)
76subrgbas 20138 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → 𝑅 = (Base‘(ℂflds 𝑅)))
87ad2antrr 723 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 = (Base‘(ℂflds 𝑅)))
95, 8sseqtrd 3972 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ (Base‘(ℂflds 𝑅)))
10 simprl 768 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ ℤ)
119, 10sseldd 3933 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ (Base‘(ℂflds 𝑅)))
12 nnz 12443 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
1312ad2antll 726 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ ℤ)
149, 13sseldd 3933 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Base‘(ℂflds 𝑅)))
15 nnne0 12108 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
1615ad2antll 726 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ 0)
17 cnfld0 20728 . . . . . . . . . . 11 0 = (0g‘ℂfld)
186, 17subrg0 20136 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → 0 = (0g‘(ℂflds 𝑅)))
1918ad2antrr 723 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 0 = (0g‘(ℂflds 𝑅)))
2016, 19neeqtrd 3010 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ (0g‘(ℂflds 𝑅)))
21 eqid 2736 . . . . . . . . . 10 (Base‘(ℂflds 𝑅)) = (Base‘(ℂflds 𝑅))
22 eqid 2736 . . . . . . . . . 10 (Unit‘(ℂflds 𝑅)) = (Unit‘(ℂflds 𝑅))
23 eqid 2736 . . . . . . . . . 10 (0g‘(ℂflds 𝑅)) = (0g‘(ℂflds 𝑅))
2421, 22, 23drngunit 20098 . . . . . . . . 9 ((ℂflds 𝑅) ∈ DivRing → (𝑦 ∈ (Unit‘(ℂflds 𝑅)) ↔ (𝑦 ∈ (Base‘(ℂflds 𝑅)) ∧ 𝑦 ≠ (0g‘(ℂflds 𝑅)))))
2524ad2antlr 724 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑦 ∈ (Unit‘(ℂflds 𝑅)) ↔ (𝑦 ∈ (Base‘(ℂflds 𝑅)) ∧ 𝑦 ≠ (0g‘(ℂflds 𝑅)))))
2614, 20, 25mpbir2and 710 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Unit‘(ℂflds 𝑅)))
27 eqid 2736 . . . . . . . 8 (/r‘(ℂflds 𝑅)) = (/r‘(ℂflds 𝑅))
2821, 22, 27dvrcl 20023 . . . . . . 7 (((ℂflds 𝑅) ∈ Ring ∧ 𝑥 ∈ (Base‘(ℂflds 𝑅)) ∧ 𝑦 ∈ (Unit‘(ℂflds 𝑅))) → (𝑥(/r‘(ℂflds 𝑅))𝑦) ∈ (Base‘(ℂflds 𝑅)))
293, 11, 26, 28syl3anc 1370 . . . . . 6 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥(/r‘(ℂflds 𝑅))𝑦) ∈ (Base‘(ℂflds 𝑅)))
30 simpll 764 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 ∈ (SubRing‘ℂfld))
315, 10sseldd 3933 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥𝑅)
32 cnflddiv 20734 . . . . . . . 8 / = (/r‘ℂfld)
336, 32, 22, 27subrgdv 20146 . . . . . . 7 ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥𝑅𝑦 ∈ (Unit‘(ℂflds 𝑅))) → (𝑥 / 𝑦) = (𝑥(/r‘(ℂflds 𝑅))𝑦))
3430, 31, 26, 33syl3anc 1370 . . . . . 6 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) = (𝑥(/r‘(ℂflds 𝑅))𝑦))
3529, 34, 83eltr4d 2852 . . . . 5 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) ∈ 𝑅)
36 eleq1 2824 . . . . 5 (𝑧 = (𝑥 / 𝑦) → (𝑧𝑅 ↔ (𝑥 / 𝑦) ∈ 𝑅))
3735, 36syl5ibrcom 246 . . . 4 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑧 = (𝑥 / 𝑦) → 𝑧𝑅))
3837rexlimdvva 3201 . . 3 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦) → 𝑧𝑅))
391, 38biimtrid 241 . 2 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → (𝑧 ∈ ℚ → 𝑧𝑅))
4039ssrdv 3938 1 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wne 2940  wrex 3070  wss 3898  cfv 6479  (class class class)co 7337  0cc0 10972   / cdiv 11733  cn 12074  cz 12420  cq 12789  Basecbs 17009  s cress 17038  0gc0g 17247  Ringcrg 19878  Unitcui 19976  /rcdvr 20019  DivRingcdr 20093  SubRingcsubrg 20125  fldccnfld 20703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-addf 11051  ax-mulf 11052
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-tpos 8112  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-5 12140  df-6 12141  df-7 12142  df-8 12143  df-9 12144  df-n0 12335  df-z 12421  df-dec 12539  df-uz 12684  df-q 12790  df-fz 13341  df-seq 13823  df-struct 16945  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-mulr 17073  df-starv 17074  df-tset 17078  df-ple 17079  df-ds 17081  df-unif 17082  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-grp 18676  df-minusg 18677  df-mulg 18797  df-subg 18848  df-cmn 19483  df-mgp 19816  df-ur 19833  df-ring 19880  df-cring 19881  df-oppr 19957  df-dvdsr 19978  df-unit 19979  df-invr 20009  df-dvr 20020  df-drng 20095  df-subrg 20127  df-cnfld 20704
This theorem is referenced by:  cphqss  24458  resscdrg  24628
  Copyright terms: Public domain W3C validator