Theorem qsssubdrg 20856

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 12875	. . 3 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
2		drngring 20192	. . . . . . . 8 ⊢ ((ℂfld ↾s 𝑅) ∈ DivRing → (ℂfld ↾s 𝑅) ∈ Ring)
3	2	ad2antlr 725	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (ℂfld ↾s 𝑅) ∈ Ring)
4		zsssubrg 20855	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
5	4	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ 𝑅)
6		eqid 2736	. . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝑅) = (ℂfld ↾s 𝑅)
7	6	subrgbas 20231	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → 𝑅 = (Base‘(ℂfld ↾s 𝑅)))
8	7	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 = (Base‘(ℂfld ↾s 𝑅)))
9	5, 8	sseqtrd 3984	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ (Base‘(ℂfld ↾s 𝑅)))
10		simprl 769	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ ℤ)
11	9, 10	sseldd 3945	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ (Base‘(ℂfld ↾s 𝑅)))
12		nnz 12520	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
13	12	ad2antll 727	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ ℤ)
14	9, 13	sseldd 3945	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Base‘(ℂfld ↾s 𝑅)))
15		nnne0 12187	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
16	15	ad2antll 727	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ 0)
17		cnfld0 20821	. . . . . . . . . . 11 ⊢ 0 = (0g‘ℂfld)
18	6, 17	subrg0 20229	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → 0 = (0g‘(ℂfld ↾s 𝑅)))
19	18	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 0 = (0g‘(ℂfld ↾s 𝑅)))
20	16, 19	neeqtrd 3013	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ (0g‘(ℂfld ↾s 𝑅)))
21		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘(ℂfld ↾s 𝑅)) = (Base‘(ℂfld ↾s 𝑅))
22		eqid 2736	. . . . . . . . . 10 ⊢ (Unit‘(ℂfld ↾s 𝑅)) = (Unit‘(ℂfld ↾s 𝑅))
23		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘(ℂfld ↾s 𝑅)) = (0g‘(ℂfld ↾s 𝑅))
24	21, 22, 23	drngunit 20190	. . . . . . . . 9 ⊢ ((ℂfld ↾s 𝑅) ∈ DivRing → (𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅)) ↔ (𝑦 ∈ (Base‘(ℂfld ↾s 𝑅)) ∧ 𝑦 ≠ (0g‘(ℂfld ↾s 𝑅)))))
25	24	ad2antlr 725	. . . . . . . 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 711	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅)))
27		eqid 2736	. . . . . . . 8 ⊢ (/r‘(ℂfld ↾s 𝑅)) = (/r‘(ℂfld ↾s 𝑅))
28	21, 22, 27	dvrcl 20115	. . . . . . 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 1371	. . . . . 6 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥(/r‘(ℂfld ↾s 𝑅))𝑦) ∈ (Base‘(ℂfld ↾s 𝑅)))
30		simpll 765	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 ∈ (SubRing‘ℂfld))
31	5, 10	sseldd 3945	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ 𝑅)
32		cnflddiv 20827	. . . . . . . 8 ⊢ / = (/r‘ℂfld)
33	6, 32, 22, 27	subrgdv 20239	. . . . . . 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 1371	. . . . . 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 2853	. . . . 5 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) ∈ 𝑅)
36		eleq1 2825	. . . . 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 3205	. . 3 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦) → 𝑧 ∈ 𝑅))
39	1, 38	biimtrid 241	. 2 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → (𝑧 ∈ ℚ → 𝑧 ∈ 𝑅))
40	39	ssrdv 3950	1 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073   ⊆ wss 3910  ‘cfv 6496  (class class class)co 7357  0cc0 11051   / cdiv 11812  ℕcn 12153  ℤcz 12499  ℚcq 12873  Basecbs 17083   ↾_s cress 17112  0_gc0g 17321  Ringcrg 19964  Unitcui 20068  /_rcdvr 20111  DivRingcdr 20185  SubRingcsubrg 20218  ℂ_fldccnfld 20796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-addf 11130  ax-mulf 11131

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-fz 13425  df-seq 13907  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-mulg 18873  df-subg 18925  df-cmn 19564  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-subrg 20220  df-cnfld 20797

This theorem is referenced by:  cphqss  24552  resscdrg  24722