Theorem qsssubdrg 20569

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) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)

Proof of Theorem qsssubdrg

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elq 12619	. . 3 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
2		drngring 19913	. . . . . . . 8 ⊢ ((ℂfld ↾s 𝑅) ∈ DivRing → (ℂfld ↾s 𝑅) ∈ Ring)
3	2	ad2antlr 723	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (ℂfld ↾s 𝑅) ∈ Ring)
4		zsssubrg 20568	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
5	4	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ 𝑅)
6		eqid 2738	. . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝑅) = (ℂfld ↾s 𝑅)
7	6	subrgbas 19948	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → 𝑅 = (Base‘(ℂfld ↾s 𝑅)))
8	7	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 = (Base‘(ℂfld ↾s 𝑅)))
9	5, 8	sseqtrd 3957	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ (Base‘(ℂfld ↾s 𝑅)))
10		simprl 767	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ ℤ)
11	9, 10	sseldd 3918	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ (Base‘(ℂfld ↾s 𝑅)))
12		nnz 12272	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
13	12	ad2antll 725	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ ℤ)
14	9, 13	sseldd 3918	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Base‘(ℂfld ↾s 𝑅)))
15		nnne0 11937	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
16	15	ad2antll 725	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ 0)
17		cnfld0 20534	. . . . . . . . . . 11 ⊢ 0 = (0g‘ℂfld)
18	6, 17	subrg0 19946	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → 0 = (0g‘(ℂfld ↾s 𝑅)))
19	18	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 0 = (0g‘(ℂfld ↾s 𝑅)))
20	16, 19	neeqtrd 3012	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ (0g‘(ℂfld ↾s 𝑅)))
21		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘(ℂfld ↾s 𝑅)) = (Base‘(ℂfld ↾s 𝑅))
22		eqid 2738	. . . . . . . . . 10 ⊢ (Unit‘(ℂfld ↾s 𝑅)) = (Unit‘(ℂfld ↾s 𝑅))
23		eqid 2738	. . . . . . . . . 10 ⊢ (0g‘(ℂfld ↾s 𝑅)) = (0g‘(ℂfld ↾s 𝑅))
24	21, 22, 23	drngunit 19911	. . . . . . . . 9 ⊢ ((ℂfld ↾s 𝑅) ∈ DivRing → (𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅)) ↔ (𝑦 ∈ (Base‘(ℂfld ↾s 𝑅)) ∧ 𝑦 ≠ (0g‘(ℂfld ↾s 𝑅)))))
25	24	ad2antlr 723	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅)) ↔ (𝑦 ∈ (Base‘(ℂfld ↾s 𝑅)) ∧ 𝑦 ≠ (0g‘(ℂfld ↾s 𝑅)))))
26	14, 20, 25	mpbir2and 709	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅)))
27		eqid 2738	. . . . . . . 8 ⊢ (/r‘(ℂfld ↾s 𝑅)) = (/r‘(ℂfld ↾s 𝑅))
28	21, 22, 27	dvrcl 19843	. . . . . . 7 ⊢ (((ℂfld ↾s 𝑅) ∈ Ring ∧ 𝑥 ∈ (Base‘(ℂfld ↾s 𝑅)) ∧ 𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅))) → (𝑥(/r‘(ℂfld ↾s 𝑅))𝑦) ∈ (Base‘(ℂfld ↾s 𝑅)))
29	3, 11, 26, 28	syl3anc 1369	. . . . . 6 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥(/r‘(ℂfld ↾s 𝑅))𝑦) ∈ (Base‘(ℂfld ↾s 𝑅)))
30		simpll 763	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 ∈ (SubRing‘ℂfld))
31	5, 10	sseldd 3918	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ 𝑅)
32		cnflddiv 20540	. . . . . . . 8 ⊢ / = (/r‘ℂfld)
33	6, 32, 22, 27	subrgdv 19956	. . . . . . 7 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅))) → (𝑥 / 𝑦) = (𝑥(/r‘(ℂfld ↾s 𝑅))𝑦))
34	30, 31, 26, 33	syl3anc 1369	. . . . . 6 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) = (𝑥(/r‘(ℂfld ↾s 𝑅))𝑦))
35	29, 34, 8	3eltr4d 2854	. . . . 5 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) ∈ 𝑅)
36		eleq1 2826	. . . . 5 ⊢ (𝑧 = (𝑥 / 𝑦) → (𝑧 ∈ 𝑅 ↔ (𝑥 / 𝑦) ∈ 𝑅))
37	35, 36	syl5ibrcom 246	. . . 4 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑧 = (𝑥 / 𝑦) → 𝑧 ∈ 𝑅))
38	37	rexlimdvva 3222	. . 3 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦) → 𝑧 ∈ 𝑅))
39	1, 38	syl5bi 241	. 2 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → (𝑧 ∈ ℚ → 𝑧 ∈ 𝑅))
40	39	ssrdv 3923	1 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   ⊆ wss 3883  ‘cfv 6418  (class class class)co 7255  0cc0 10802   / cdiv 11562  ℕcn 11903  ℤcz 12249  ℚcq 12617  Basecbs 16840   ↾_s cress 16867  0_gc0g 17067  Ringcrg 19698  Unitcui 19796  /_rcdvr 19839  DivRingcdr 19906  SubRingcsubrg 19935  ℂ_fldccnfld 20510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-addf 10881  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-fz 13169  df-seq 13650  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-mulg 18616  df-subg 18667  df-cmn 19303  df-mgp 19636  df-ur 19653  df-ring 19700  df-cring 19701  df-oppr 19777  df-dvdsr 19798  df-unit 19799  df-invr 19829  df-dvr 19840  df-drng 19908  df-subrg 19937  df-cnfld 20511

This theorem is referenced by:  cphqss  24257  resscdrg  24427