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Theorem qsssubdrg 20020
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 11993 . . 3 (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
2 drngring 18964 . . . . . . . 8 ((ℂflds 𝑅) ∈ DivRing → (ℂflds 𝑅) ∈ Ring)
32ad2antlr 706 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (ℂflds 𝑅) ∈ Ring)
4 zsssubrg 20019 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
54ad2antrr 705 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ 𝑅)
6 eqid 2771 . . . . . . . . . . 11 (ℂflds 𝑅) = (ℂflds 𝑅)
76subrgbas 18999 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → 𝑅 = (Base‘(ℂflds 𝑅)))
87ad2antrr 705 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 = (Base‘(ℂflds 𝑅)))
95, 8sseqtrd 3790 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ (Base‘(ℂflds 𝑅)))
10 simprl 754 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ ℤ)
119, 10sseldd 3753 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ (Base‘(ℂflds 𝑅)))
12 nnz 11601 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
1312ad2antll 708 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ ℤ)
149, 13sseldd 3753 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Base‘(ℂflds 𝑅)))
15 nnne0 11255 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
1615ad2antll 708 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ 0)
17 cnfld0 19985 . . . . . . . . . . 11 0 = (0g‘ℂfld)
186, 17subrg0 18997 . . . . . . . . . 10 (𝑅 ∈ (SubRing‘ℂfld) → 0 = (0g‘(ℂflds 𝑅)))
1918ad2antrr 705 . . . . . . . . 9 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 0 = (0g‘(ℂflds 𝑅)))
2016, 19neeqtrd 3012 . . . . . . . 8 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ (0g‘(ℂflds 𝑅)))
21 eqid 2771 . . . . . . . . . 10 (Base‘(ℂflds 𝑅)) = (Base‘(ℂflds 𝑅))
22 eqid 2771 . . . . . . . . . 10 (Unit‘(ℂflds 𝑅)) = (Unit‘(ℂflds 𝑅))
23 eqid 2771 . . . . . . . . . 10 (0g‘(ℂflds 𝑅)) = (0g‘(ℂflds 𝑅))
2421, 22, 23drngunit 18962 . . . . . . . . 9 ((ℂflds 𝑅) ∈ DivRing → (𝑦 ∈ (Unit‘(ℂflds 𝑅)) ↔ (𝑦 ∈ (Base‘(ℂflds 𝑅)) ∧ 𝑦 ≠ (0g‘(ℂflds 𝑅)))))
2524ad2antlr 706 . . . . . . . 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 692 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Unit‘(ℂflds 𝑅)))
27 eqid 2771 . . . . . . . 8 (/r‘(ℂflds 𝑅)) = (/r‘(ℂflds 𝑅))
2821, 22, 27dvrcl 18894 . . . . . . 7 (((ℂflds 𝑅) ∈ Ring ∧ 𝑥 ∈ (Base‘(ℂflds 𝑅)) ∧ 𝑦 ∈ (Unit‘(ℂflds 𝑅))) → (𝑥(/r‘(ℂflds 𝑅))𝑦) ∈ (Base‘(ℂflds 𝑅)))
293, 11, 26, 28syl3anc 1476 . . . . . 6 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥(/r‘(ℂflds 𝑅))𝑦) ∈ (Base‘(ℂflds 𝑅)))
30 simpll 750 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 ∈ (SubRing‘ℂfld))
315, 10sseldd 3753 . . . . . . 7 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥𝑅)
32 cnflddiv 19991 . . . . . . . 8 / = (/r‘ℂfld)
336, 32, 22, 27subrgdv 19007 . . . . . . 7 ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥𝑅𝑦 ∈ (Unit‘(ℂflds 𝑅))) → (𝑥 / 𝑦) = (𝑥(/r‘(ℂflds 𝑅))𝑦))
3430, 31, 26, 33syl3anc 1476 . . . . . 6 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) = (𝑥(/r‘(ℂflds 𝑅))𝑦))
3529, 34, 83eltr4d 2865 . . . . 5 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) ∈ 𝑅)
36 eleq1 2838 . . . . 5 (𝑧 = (𝑥 / 𝑦) → (𝑧𝑅 ↔ (𝑥 / 𝑦) ∈ 𝑅))
3735, 36syl5ibrcom 237 . . . 4 (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑧 = (𝑥 / 𝑦) → 𝑧𝑅))
3837rexlimdvva 3186 . . 3 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦) → 𝑧𝑅))
391, 38syl5bi 232 . 2 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → (𝑧 ∈ ℚ → 𝑧𝑅))
4039ssrdv 3758 1 ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wrex 3062  wss 3723  cfv 6031  (class class class)co 6793  0cc0 10138   / cdiv 10886  cn 11222  cz 11579  cq 11991  Basecbs 16064  s cress 16065  0gc0g 16308  Ringcrg 18755  Unitcui 18847  /rcdvr 18890  DivRingcdr 18957  SubRingcsubrg 18986  fldccnfld 19961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-addf 10217  ax-mulf 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-tpos 7504  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-q 11992  df-fz 12534  df-seq 13009  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-tset 16168  df-ple 16169  df-ds 16172  df-unif 16173  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-mulg 17749  df-subg 17799  df-cmn 18402  df-mgp 18698  df-ur 18710  df-ring 18757  df-cring 18758  df-oppr 18831  df-dvdsr 18849  df-unit 18850  df-invr 18880  df-dvr 18891  df-drng 18959  df-subrg 18988  df-cnfld 19962
This theorem is referenced by:  cphqss  23207  resscdrg  23373
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