Theorem qsssubdrg 20394

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 12529	. . 3 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
2		drngring 19746	. . . . . . . 8 ⊢ ((ℂfld ↾s 𝑅) ∈ DivRing → (ℂfld ↾s 𝑅) ∈ Ring)
3	2	ad2antlr 727	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (ℂfld ↾s 𝑅) ∈ Ring)
4		zsssubrg 20393	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
5	4	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ 𝑅)
6		eqid 2734	. . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝑅) = (ℂfld ↾s 𝑅)
7	6	subrgbas 19781	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → 𝑅 = (Base‘(ℂfld ↾s 𝑅)))
8	7	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 = (Base‘(ℂfld ↾s 𝑅)))
9	5, 8	sseqtrd 3931	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ (Base‘(ℂfld ↾s 𝑅)))
10		simprl 771	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ ℤ)
11	9, 10	sseldd 3892	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ (Base‘(ℂfld ↾s 𝑅)))
12		nnz 12182	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
13	12	ad2antll 729	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ ℤ)
14	9, 13	sseldd 3892	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Base‘(ℂfld ↾s 𝑅)))
15		nnne0 11847	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
16	15	ad2antll 729	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ 0)
17		cnfld0 20359	. . . . . . . . . . 11 ⊢ 0 = (0g‘ℂfld)
18	6, 17	subrg0 19779	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → 0 = (0g‘(ℂfld ↾s 𝑅)))
19	18	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 0 = (0g‘(ℂfld ↾s 𝑅)))
20	16, 19	neeqtrd 3004	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ (0g‘(ℂfld ↾s 𝑅)))
21		eqid 2734	. . . . . . . . . 10 ⊢ (Base‘(ℂfld ↾s 𝑅)) = (Base‘(ℂfld ↾s 𝑅))
22		eqid 2734	. . . . . . . . . 10 ⊢ (Unit‘(ℂfld ↾s 𝑅)) = (Unit‘(ℂfld ↾s 𝑅))
23		eqid 2734	. . . . . . . . . 10 ⊢ (0g‘(ℂfld ↾s 𝑅)) = (0g‘(ℂfld ↾s 𝑅))
24	21, 22, 23	drngunit 19744	. . . . . . . . 9 ⊢ ((ℂfld ↾s 𝑅) ∈ DivRing → (𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅)) ↔ (𝑦 ∈ (Base‘(ℂfld ↾s 𝑅)) ∧ 𝑦 ≠ (0g‘(ℂfld ↾s 𝑅)))))
25	24	ad2antlr 727	. . . . . . . 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 713	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅)))
27		eqid 2734	. . . . . . . 8 ⊢ (/r‘(ℂfld ↾s 𝑅)) = (/r‘(ℂfld ↾s 𝑅))
28	21, 22, 27	dvrcl 19676	. . . . . . 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 1373	. . . . . 6 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥(/r‘(ℂfld ↾s 𝑅))𝑦) ∈ (Base‘(ℂfld ↾s 𝑅)))
30		simpll 767	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 ∈ (SubRing‘ℂfld))
31	5, 10	sseldd 3892	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ 𝑅)
32		cnflddiv 20365	. . . . . . . 8 ⊢ / = (/r‘ℂfld)
33	6, 32, 22, 27	subrgdv 19789	. . . . . . 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 1373	. . . . . 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 2849	. . . . 5 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) ∈ 𝑅)
36		eleq1 2821	. . . . 5 ⊢ (𝑧 = (𝑥 / 𝑦) → (𝑧 ∈ 𝑅 ↔ (𝑥 / 𝑦) ∈ 𝑅))
37	35, 36	syl5ibrcom 250	. . . 4 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑧 = (𝑥 / 𝑦) → 𝑧 ∈ 𝑅))
38	37	rexlimdvva 3206	. . 3 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦) → 𝑧 ∈ 𝑅))
39	1, 38	syl5bi 245	. 2 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → (𝑧 ∈ ℚ → 𝑧 ∈ 𝑅))
40	39	ssrdv 3897	1 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2935  ∃wrex 3055   ⊆ wss 3857  ‘cfv 6369  (class class class)co 7202  0cc0 10712   / cdiv 11472  ℕcn 11813  ℤcz 12159  ℚcq 12527  Basecbs 16684   ↾_s cress 16685  0_gc0g 16916  Ringcrg 19534  Unitcui 19629  /_rcdvr 19672  DivRingcdr 19739  SubRingcsubrg 19768  ℂ_fldccnfld 20335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789  ax-addf 10791  ax-mulf 10792

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-tpos 7957  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-er 8380  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-div 11473  df-nn 11814  df-2 11876  df-3 11877  df-4 11878  df-5 11879  df-6 11880  df-7 11881  df-8 11882  df-9 11883  df-n0 12074  df-z 12160  df-dec 12277  df-uz 12422  df-q 12528  df-fz 13079  df-seq 13558  df-struct 16686  df-ndx 16687  df-slot 16688  df-base 16690  df-sets 16691  df-ress 16692  df-plusg 16780  df-mulr 16781  df-starv 16782  df-tset 16786  df-ple 16787  df-ds 16789  df-unif 16790  df-0g 16918  df-mgm 18086  df-sgrp 18135  df-mnd 18146  df-grp 18340  df-minusg 18341  df-mulg 18461  df-subg 18512  df-cmn 19144  df-mgp 19477  df-ur 19489  df-ring 19536  df-cring 19537  df-oppr 19613  df-dvdsr 19631  df-unit 19632  df-invr 19662  df-dvr 19673  df-drng 19741  df-subrg 19770  df-cnfld 20336

This theorem is referenced by:  cphqss  24057  resscdrg  24227