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Theorem qsssubdrg 20872
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 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) β†’ β„š βŠ† 𝑅)

Proof of Theorem qsssubdrg
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elq 12882 . . 3 (𝑧 ∈ β„š ↔ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„• 𝑧 = (π‘₯ / 𝑦))
2 drngring 20206 . . . . . . . 8 ((β„‚fld β†Ύs 𝑅) ∈ DivRing β†’ (β„‚fld β†Ύs 𝑅) ∈ Ring)
32ad2antlr 726 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (β„‚fld β†Ύs 𝑅) ∈ Ring)
4 zsssubrg 20871 . . . . . . . . . 10 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ β„€ βŠ† 𝑅)
54ad2antrr 725 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ β„€ βŠ† 𝑅)
6 eqid 2737 . . . . . . . . . . 11 (β„‚fld β†Ύs 𝑅) = (β„‚fld β†Ύs 𝑅)
76subrgbas 20247 . . . . . . . . . 10 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ 𝑅 = (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
87ad2antrr 725 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑅 = (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
95, 8sseqtrd 3989 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ β„€ βŠ† (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
10 simprl 770 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ π‘₯ ∈ β„€)
119, 10sseldd 3950 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ π‘₯ ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
12 nnz 12527 . . . . . . . . . 10 (𝑦 ∈ β„• β†’ 𝑦 ∈ β„€)
1312ad2antll 728 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 ∈ β„€)
149, 13sseldd 3950 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
15 nnne0 12194 . . . . . . . . . 10 (𝑦 ∈ β„• β†’ 𝑦 β‰  0)
1615ad2antll 728 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 β‰  0)
17 cnfld0 20837 . . . . . . . . . . 11 0 = (0gβ€˜β„‚fld)
186, 17subrg0 20245 . . . . . . . . . 10 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ 0 = (0gβ€˜(β„‚fld β†Ύs 𝑅)))
1918ad2antrr 725 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 0 = (0gβ€˜(β„‚fld β†Ύs 𝑅)))
2016, 19neeqtrd 3014 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 β‰  (0gβ€˜(β„‚fld β†Ύs 𝑅)))
21 eqid 2737 . . . . . . . . . 10 (Baseβ€˜(β„‚fld β†Ύs 𝑅)) = (Baseβ€˜(β„‚fld β†Ύs 𝑅))
22 eqid 2737 . . . . . . . . . 10 (Unitβ€˜(β„‚fld β†Ύs 𝑅)) = (Unitβ€˜(β„‚fld β†Ύs 𝑅))
23 eqid 2737 . . . . . . . . . 10 (0gβ€˜(β„‚fld β†Ύs 𝑅)) = (0gβ€˜(β„‚fld β†Ύs 𝑅))
2421, 22, 23drngunit 20204 . . . . . . . . 9 ((β„‚fld β†Ύs 𝑅) ∈ DivRing β†’ (𝑦 ∈ (Unitβ€˜(β„‚fld β†Ύs 𝑅)) ↔ (𝑦 ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)) ∧ 𝑦 β‰  (0gβ€˜(β„‚fld β†Ύs 𝑅)))))
2524ad2antlr 726 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (𝑦 ∈ (Unitβ€˜(β„‚fld β†Ύs 𝑅)) ↔ (𝑦 ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)) ∧ 𝑦 β‰  (0gβ€˜(β„‚fld β†Ύs 𝑅)))))
2614, 20, 25mpbir2and 712 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 ∈ (Unitβ€˜(β„‚fld β†Ύs 𝑅)))
27 eqid 2737 . . . . . . . 8 (/rβ€˜(β„‚fld β†Ύs 𝑅)) = (/rβ€˜(β„‚fld β†Ύs 𝑅))
2821, 22, 27dvrcl 20122 . . . . . . 7 (((β„‚fld β†Ύs 𝑅) ∈ Ring ∧ π‘₯ ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)) ∧ 𝑦 ∈ (Unitβ€˜(β„‚fld β†Ύs 𝑅))) β†’ (π‘₯(/rβ€˜(β„‚fld β†Ύs 𝑅))𝑦) ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
293, 11, 26, 28syl3anc 1372 . . . . . 6 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (π‘₯(/rβ€˜(β„‚fld β†Ύs 𝑅))𝑦) ∈ (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
30 simpll 766 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑅 ∈ (SubRingβ€˜β„‚fld))
315, 10sseldd 3950 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ π‘₯ ∈ 𝑅)
32 cnflddiv 20843 . . . . . . . 8 / = (/rβ€˜β„‚fld)
336, 32, 22, 27subrgdv 20255 . . . . . . 7 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ π‘₯ ∈ 𝑅 ∧ 𝑦 ∈ (Unitβ€˜(β„‚fld β†Ύs 𝑅))) β†’ (π‘₯ / 𝑦) = (π‘₯(/rβ€˜(β„‚fld β†Ύs 𝑅))𝑦))
3430, 31, 26, 33syl3anc 1372 . . . . . 6 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (π‘₯ / 𝑦) = (π‘₯(/rβ€˜(β„‚fld β†Ύs 𝑅))𝑦))
3529, 34, 83eltr4d 2853 . . . . 5 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (π‘₯ / 𝑦) ∈ 𝑅)
36 eleq1 2826 . . . . 5 (𝑧 = (π‘₯ / 𝑦) β†’ (𝑧 ∈ 𝑅 ↔ (π‘₯ / 𝑦) ∈ 𝑅))
3735, 36syl5ibrcom 247 . . . 4 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (𝑧 = (π‘₯ / 𝑦) β†’ 𝑧 ∈ 𝑅))
3837rexlimdvva 3206 . . 3 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) β†’ (βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„• 𝑧 = (π‘₯ / 𝑦) β†’ 𝑧 ∈ 𝑅))
391, 38biimtrid 241 . 2 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) β†’ (𝑧 ∈ β„š β†’ 𝑧 ∈ 𝑅))
4039ssrdv 3955 1 ((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) β†’ β„š βŠ† 𝑅)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074   βŠ† wss 3915  β€˜cfv 6501  (class class class)co 7362  0cc0 11058   / cdiv 11819  β„•cn 12160  β„€cz 12506  β„šcq 12880  Basecbs 17090   β†Ύs cress 17119  0gc0g 17328  Ringcrg 19971  Unitcui 20075  /rcdvr 20118  DivRingcdr 20199  SubRingcsubrg 20234  β„‚fldccnfld 20812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-addf 11137  ax-mulf 11138
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-tpos 8162  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-q 12881  df-fz 13432  df-seq 13914  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-starv 17155  df-tset 17159  df-ple 17160  df-ds 17162  df-unif 17163  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-mulg 18880  df-subg 18932  df-cmn 19571  df-mgp 19904  df-ur 19921  df-ring 19973  df-cring 19974  df-oppr 20056  df-dvdsr 20077  df-unit 20078  df-invr 20108  df-dvr 20119  df-drng 20201  df-subrg 20236  df-cnfld 20813
This theorem is referenced by:  cphqss  24568  resscdrg  24738
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