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Theorem qsssubdrg 20154
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 12342 . . 3 (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
2 drngring 19506 . . . . . . . 8 ((ℂflds 𝑅) ∈ DivRing → (ℂflds 𝑅) ∈ Ring)
32ad2antlr 726 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (ℂflds 𝑅) ∈ Ring)
4 zsssubrg 20153 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
54ad2antrr 725 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ 𝑅)
6 eqid 2801 . . . . . . . . . . 11 (ℂflds 𝑅) = (ℂflds 𝑅)
76subrgbas 19541 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → 𝑅 = (Base‘(ℂflds 𝑅)))
87ad2antrr 725 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 = (Base‘(ℂflds 𝑅)))
95, 8sseqtrd 3958 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ (Base‘(ℂflds 𝑅)))
10 simprl 770 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ ℤ)
119, 10sseldd 3919 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ (Base‘(ℂflds 𝑅)))
12 nnz 11996 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
1312ad2antll 728 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ ℤ)
149, 13sseldd 3919 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Base‘(ℂflds 𝑅)))
15 nnne0 11663 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
1615ad2antll 728 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ 0)
17 cnfld0 20119 . . . . . . . . . . 11 0 = (0g‘ℂfld)
186, 17subrg0 19539 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → 0 = (0g‘(ℂflds 𝑅)))
1918ad2antrr 725 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 0 = (0g‘(ℂflds 𝑅)))
2016, 19neeqtrd 3059 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ (0g‘(ℂflds 𝑅)))
21 eqid 2801 . . . . . . . . . 10 (Base‘(ℂflds 𝑅)) = (Base‘(ℂflds 𝑅))
22 eqid 2801 . . . . . . . . . 10 (Unit‘(ℂflds 𝑅)) = (Unit‘(ℂflds 𝑅))
23 eqid 2801 . . . . . . . . . 10 (0g‘(ℂflds 𝑅)) = (0g‘(ℂflds 𝑅))
2421, 22, 23drngunit 19504 . . . . . . . . 9 ((ℂflds 𝑅) ∈ DivRing → (𝑦 ∈ (Unit‘(ℂflds 𝑅)) ↔ (𝑦 ∈ (Base‘(ℂflds 𝑅)) ∧ 𝑦 ≠ (0g‘(ℂflds 𝑅)))))
2524ad2antlr 726 . . . . . . . 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 712 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Unit‘(ℂflds 𝑅)))
27 eqid 2801 . . . . . . . 8 (/r‘(ℂflds 𝑅)) = (/r‘(ℂflds 𝑅))
2821, 22, 27dvrcl 19436 . . . . . . 7 (((ℂflds 𝑅) ∈ Ring ∧ 𝑥 ∈ (Base‘(ℂflds 𝑅)) ∧ 𝑦 ∈ (Unit‘(ℂflds 𝑅))) → (𝑥(/r‘(ℂflds 𝑅))𝑦) ∈ (Base‘(ℂflds 𝑅)))
293, 11, 26, 28syl3anc 1368 . . . . . 6 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥(/r‘(ℂflds 𝑅))𝑦) ∈ (Base‘(ℂflds 𝑅)))
30 simpll 766 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 ∈ (SubRing‘ℂfld))
315, 10sseldd 3919 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥𝑅)
32 cnflddiv 20125 . . . . . . . 8 / = (/r‘ℂfld)
336, 32, 22, 27subrgdv 19549 . . . . . . 7 ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥𝑅𝑦 ∈ (Unit‘(ℂflds 𝑅))) → (𝑥 / 𝑦) = (𝑥(/r‘(ℂflds 𝑅))𝑦))
3430, 31, 26, 33syl3anc 1368 . . . . . 6 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) = (𝑥(/r‘(ℂflds 𝑅))𝑦))
3529, 34, 83eltr4d 2908 . . . . 5 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) ∈ 𝑅)
36 eleq1 2880 . . . . 5 (𝑧 = (𝑥 / 𝑦) → (𝑧𝑅 ↔ (𝑥 / 𝑦) ∈ 𝑅))
3735, 36syl5ibrcom 250 . . . 4 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑧 = (𝑥 / 𝑦) → 𝑧𝑅))
3837rexlimdvva 3256 . . 3 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦) → 𝑧𝑅))
391, 38syl5bi 245 . 2 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → (𝑧 ∈ ℚ → 𝑧𝑅))
4039ssrdv 3924 1 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wrex 3110  wss 3884  cfv 6328  (class class class)co 7139  0cc0 10530   / cdiv 11290  cn 11629  cz 11973  cq 12340  Basecbs 16479  s cress 16480  0gc0g 16709  Ringcrg 19294  Unitcui 19389  /rcdvr 19432  DivRingcdr 19499  SubRingcsubrg 19528  fldccnfld 20095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-addf 10609  ax-mulf 10610
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-fz 12890  df-seq 13369  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-mulr 16575  df-starv 16576  df-tset 16580  df-ple 16581  df-ds 16583  df-unif 16584  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-grp 18102  df-minusg 18103  df-mulg 18221  df-subg 18272  df-cmn 18904  df-mgp 19237  df-ur 19249  df-ring 19296  df-cring 19297  df-oppr 19373  df-dvdsr 19391  df-unit 19392  df-invr 19422  df-dvr 19433  df-drng 19501  df-subrg 19530  df-cnfld 20096
This theorem is referenced by:  cphqss  23797  resscdrg  23966
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