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Theorem qsssubdrg 19624
 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 11666 . . 3 (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
2 drngring 18577 . . . . . . . 8 ((ℂflds 𝑅) ∈ DivRing → (ℂflds 𝑅) ∈ Ring)
32ad2antlr 759 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (ℂflds 𝑅) ∈ Ring)
4 zsssubrg 19623 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
54ad2antrr 758 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ 𝑅)
6 eqid 2610 . . . . . . . . . . 11 (ℂflds 𝑅) = (ℂflds 𝑅)
76subrgbas 18612 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → 𝑅 = (Base‘(ℂflds 𝑅)))
87ad2antrr 758 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 = (Base‘(ℂflds 𝑅)))
95, 8sseqtrd 3604 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ (Base‘(ℂflds 𝑅)))
10 simprl 790 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ ℤ)
119, 10sseldd 3569 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ (Base‘(ℂflds 𝑅)))
12 nnz 11276 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
1312ad2antll 761 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ ℤ)
149, 13sseldd 3569 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Base‘(ℂflds 𝑅)))
15 nnne0 10930 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
1615ad2antll 761 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ 0)
17 cnfld0 19589 . . . . . . . . . . 11 0 = (0g‘ℂfld)
186, 17subrg0 18610 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → 0 = (0g‘(ℂflds 𝑅)))
1918ad2antrr 758 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 0 = (0g‘(ℂflds 𝑅)))
2016, 19neeqtrd 2851 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ (0g‘(ℂflds 𝑅)))
21 eqid 2610 . . . . . . . . . 10 (Base‘(ℂflds 𝑅)) = (Base‘(ℂflds 𝑅))
22 eqid 2610 . . . . . . . . . 10 (Unit‘(ℂflds 𝑅)) = (Unit‘(ℂflds 𝑅))
23 eqid 2610 . . . . . . . . . 10 (0g‘(ℂflds 𝑅)) = (0g‘(ℂflds 𝑅))
2421, 22, 23drngunit 18575 . . . . . . . . 9 ((ℂflds 𝑅) ∈ DivRing → (𝑦 ∈ (Unit‘(ℂflds 𝑅)) ↔ (𝑦 ∈ (Base‘(ℂflds 𝑅)) ∧ 𝑦 ≠ (0g‘(ℂflds 𝑅)))))
2524ad2antlr 759 . . . . . . . 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 959 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Unit‘(ℂflds 𝑅)))
27 eqid 2610 . . . . . . . 8 (/r‘(ℂflds 𝑅)) = (/r‘(ℂflds 𝑅))
2821, 22, 27dvrcl 18509 . . . . . . 7 (((ℂflds 𝑅) ∈ Ring ∧ 𝑥 ∈ (Base‘(ℂflds 𝑅)) ∧ 𝑦 ∈ (Unit‘(ℂflds 𝑅))) → (𝑥(/r‘(ℂflds 𝑅))𝑦) ∈ (Base‘(ℂflds 𝑅)))
293, 11, 26, 28syl3anc 1318 . . . . . 6 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥(/r‘(ℂflds 𝑅))𝑦) ∈ (Base‘(ℂflds 𝑅)))
30 simpll 786 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 ∈ (SubRing‘ℂfld))
315, 10sseldd 3569 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥𝑅)
32 cnflddiv 19595 . . . . . . . 8 / = (/r‘ℂfld)
336, 32, 22, 27subrgdv 18620 . . . . . . 7 ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥𝑅𝑦 ∈ (Unit‘(ℂflds 𝑅))) → (𝑥 / 𝑦) = (𝑥(/r‘(ℂflds 𝑅))𝑦))
3430, 31, 26, 33syl3anc 1318 . . . . . 6 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) = (𝑥(/r‘(ℂflds 𝑅))𝑦))
3529, 34, 83eltr4d 2703 . . . . 5 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) ∈ 𝑅)
36 eleq1 2676 . . . . 5 (𝑧 = (𝑥 / 𝑦) → (𝑧𝑅 ↔ (𝑥 / 𝑦) ∈ 𝑅))
3735, 36syl5ibrcom 236 . . . 4 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑧 = (𝑥 / 𝑦) → 𝑧𝑅))
3837rexlimdvva 3020 . . 3 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦) → 𝑧𝑅))
391, 38syl5bi 231 . 2 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → (𝑧 ∈ ℚ → 𝑧𝑅))
4039ssrdv 3574 1 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  0cc0 9815   / cdiv 10563  ℕcn 10897  ℤcz 11254  ℚcq 11664  Basecbs 15695   ↾s cress 15696  0gc0g 15923  Ringcrg 18370  Unitcui 18462  /rcdvr 18505  DivRingcdr 18570  SubRingcsubrg 18599  ℂfldccnfld 19567 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-fz 12198  df-seq 12664  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-mulg 17364  df-subg 17414  df-cmn 18018  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-subrg 18601  df-cnfld 19568 This theorem is referenced by:  cphqss  22796  resscdrg  22962
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