Theorem qsssubdrg 21466

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 12945	. . 3 ⊢ (𝑧 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦))
2		drngring 20773	. . . . . . . 8 ⊢ ((ℂfld ↾s 𝑅) ∈ DivRing → (ℂfld ↾s 𝑅) ∈ Ring)
3	2	ad2antlr 737	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (ℂfld ↾s 𝑅) ∈ Ring)
4		zsssubrg 21465	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
5	4	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ 𝑅)
6		eqid 2761	. . . . . . . . . . 11 ⊢ (ℂfld ↾s 𝑅) = (ℂfld ↾s 𝑅)
7	6	subrgbas 20618	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → 𝑅 = (Base‘(ℂfld ↾s 𝑅)))
8	7	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 = (Base‘(ℂfld ↾s 𝑅)))
9	5, 8	sseqtrd 3970	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → ℤ ⊆ (Base‘(ℂfld ↾s 𝑅)))
10		simprl 780	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ ℤ)
11	9, 10	sseldd 3935	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ (Base‘(ℂfld ↾s 𝑅)))
12		nnz 12583	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
13	12	ad2antll 739	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ ℤ)
14	9, 13	sseldd 3935	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Base‘(ℂfld ↾s 𝑅)))
15		nnne0 12241	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
16	15	ad2antll 739	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ 0)
17		cnfld0 21436	. . . . . . . . . . 11 ⊢ 0 = (0g‘ℂfld)
18	6, 17	subrg0 20616	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → 0 = (0g‘(ℂfld ↾s 𝑅)))
19	18	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 0 = (0g‘(ℂfld ↾s 𝑅)))
20	16, 19	neeqtrd 3025	. . . . . . . 8 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ≠ (0g‘(ℂfld ↾s 𝑅)))
21		eqid 2761	. . . . . . . . . 10 ⊢ (Base‘(ℂfld ↾s 𝑅)) = (Base‘(ℂfld ↾s 𝑅))
22		eqid 2761	. . . . . . . . . 10 ⊢ (Unit‘(ℂfld ↾s 𝑅)) = (Unit‘(ℂfld ↾s 𝑅))
23		eqid 2761	. . . . . . . . . 10 ⊢ (0g‘(ℂfld ↾s 𝑅)) = (0g‘(ℂfld ↾s 𝑅))
24	21, 22, 23	drngunit 20771	. . . . . . . . 9 ⊢ ((ℂfld ↾s 𝑅) ∈ DivRing → (𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅)) ↔ (𝑦 ∈ (Base‘(ℂfld ↾s 𝑅)) ∧ 𝑦 ≠ (0g‘(ℂfld ↾s 𝑅)))))
25	24	ad2antlr 737	. . . . . . . 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 723	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑦 ∈ (Unit‘(ℂfld ↾s 𝑅)))
27		eqid 2761	. . . . . . . 8 ⊢ (/r‘(ℂfld ↾s 𝑅)) = (/r‘(ℂfld ↾s 𝑅))
28	21, 22, 27	dvrcl 20440	. . . . . . 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 1389	. . . . . 6 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥(/r‘(ℂfld ↾s 𝑅))𝑦) ∈ (Base‘(ℂfld ↾s 𝑅)))
30		simpll 776	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑅 ∈ (SubRing‘ℂfld))
31	5, 10	sseldd 3935	. . . . . . 7 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → 𝑥 ∈ 𝑅)
32		cnflddiv 21442	. . . . . . . 8 ⊢ / = (/r‘ℂfld)
33	6, 32, 22, 27	subrgdv 20626	. . . . . . 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 1389	. . . . . 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 2876	. . . . 5 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑥 / 𝑦) ∈ 𝑅)
36		eleq1 2849	. . . . 5 ⊢ (𝑧 = (𝑥 / 𝑦) → (𝑧 ∈ 𝑅 ↔ (𝑥 / 𝑦) ∈ 𝑅))
37	35, 36	syl5ibrcom 249	. . . 4 ⊢ (((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ)) → (𝑧 = (𝑥 / 𝑦) → 𝑧 ∈ 𝑅))
38	37	rexlimdvva 3218	. . 3 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑧 = (𝑥 / 𝑦) → 𝑧 ∈ 𝑅))
39	1, 38	biimtrid 244	. 2 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → (𝑧 ∈ ℚ → 𝑧 ∈ 𝑅))
40	39	ssrdv 3940	1 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085   ⊆ wss 3902  ‘cfv 6516  (class class class)co 7391  0cc0 11067   / cdiv 11838  ℕcn 12204  ℤcz 12562  ℚcq 12943  Basecbs 17236   ↾_s cress 17257  0_gc0g 17459  Ringcrg 20270  Unitcui 20391  /_rcdvr 20436  SubRingcsubrg 20606  DivRingcdr 20766  ℂ_fldccnfld 21412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144  ax-addf 11146

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-tpos 8200  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-er 8672  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-div 11839  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-z 12563  df-dec 12683  df-uz 12834  df-q 12944  df-fz 13507  df-seq 14009  df-struct 17174  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-ress 17258  df-plusg 17290  df-mulr 17291  df-starv 17292  df-tset 17296  df-ple 17297  df-ds 17299  df-unif 17300  df-0g 17461  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18969  df-minusg 18970  df-mulg 19101  df-subg 19156  df-cmn 19813  df-abl 19814  df-mgp 20178  df-rng 20190  df-ur 20219  df-ring 20272  df-cring 20273  df-oppr 20373  df-dvdsr 20393  df-unit 20394  df-invr 20424  df-dvr 20437  df-subrg 20607  df-drng 20768  df-cnfld 21413

This theorem is referenced by:  cphqss  25238  resscdrg  25408