Theorem qsssubdrg 21392

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 12964	. . 3 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
2		drngring 20694	. . . . . . . 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 21391	. . . . . . . . . 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 2735	. . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝑅) = (ℂfld ↾s 𝑅)
7	6	subrgbas 20539	. . . . . . . . . 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 3995	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ (Base‘(ℂfld ↾s 𝑅)))
10		simprl 770	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ ℤ)
11	9, 10	sseldd 3959	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ (Base‘(ℂfld ↾s 𝑅)))
12		nnz 12607	. . . . . . . . . 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 3959	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Base‘(ℂfld ↾s 𝑅)))
15		nnne0 12272	. . . . . . . . . 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 21353	. . . . . . . . . . 11 ⊢ 0 = (0g‘ℂfld)
18	6, 17	subrg0 20537	. . . . . . . . . 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 3001	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ (0g‘(ℂfld ↾s 𝑅)))
21		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘(ℂfld ↾s 𝑅)) = (Base‘(ℂfld ↾s 𝑅))
22		eqid 2735	. . . . . . . . . 10 ⊢ (Unit‘(ℂfld ↾s 𝑅)) = (Unit‘(ℂfld ↾s 𝑅))
23		eqid 2735	. . . . . . . . . 10 ⊢ (0g‘(ℂfld ↾s 𝑅)) = (0g‘(ℂfld ↾s 𝑅))
24	21, 22, 23	drngunit 20692	. . . . . . . . 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 2735	. . . . . . . 8 ⊢ (/r‘(ℂfld ↾s 𝑅)) = (/r‘(ℂfld ↾s 𝑅))
28	21, 22, 27	dvrcl 20362	. . . . . . 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 766	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 ∈ (SubRing‘ℂfld))
31	5, 10	sseldd 3959	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ 𝑅)
32		cnflddiv 21361	. . . . . . . 8 ⊢ / = (/r‘ℂfld)
33	6, 32, 22, 27	subrgdv 20547	. . . . . . 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 2822	. . . . 5 ⊢ (𝑧 = (𝑥 / 𝑦) → (𝑧 ∈ 𝑅 ↔ (𝑥 / 𝑦) ∈ 𝑅))
37	35, 36	syl5ibrcom 247	. . . 4 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑧 = (𝑥 / 𝑦) → 𝑧 ∈ 𝑅))
38	37	rexlimdvva 3198	. . 3 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦) → 𝑧 ∈ 𝑅))
39	1, 38	biimtrid 242	. 2 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → (𝑧 ∈ ℚ → 𝑧 ∈ 𝑅))
40	39	ssrdv 3964	1 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∃wrex 3060   ⊆ wss 3926  ‘cfv 6530  (class class class)co 7403  0cc0 11127   / cdiv 11892  ℕcn 12238  ℤcz 12586  ℚcq 12962  Basecbs 17226   ↾_s cress 17249  0_gc0g 17451  Ringcrg 20191  Unitcui 20313  /_rcdvr 20358  SubRingcsubrg 20527  DivRingcdr 20687  ℂ_fldccnfld 21313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-addf 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-tpos 8223  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12851  df-q 12963  df-fz 13523  df-seq 14018  df-struct 17164  df-sets 17181  df-slot 17199  df-ndx 17211  df-base 17227  df-ress 17250  df-plusg 17282  df-mulr 17283  df-starv 17284  df-tset 17288  df-ple 17289  df-ds 17291  df-unif 17292  df-0g 17453  df-mgm 18616  df-sgrp 18695  df-mnd 18711  df-grp 18917  df-minusg 18918  df-mulg 19049  df-subg 19104  df-cmn 19761  df-abl 19762  df-mgp 20099  df-rng 20111  df-ur 20140  df-ring 20193  df-cring 20194  df-oppr 20295  df-dvdsr 20315  df-unit 20316  df-invr 20346  df-dvr 20359  df-subrg 20528  df-drng 20689  df-cnfld 21314

This theorem is referenced by:  cphqss  25138  resscdrg  25308