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Theorem qsssubdrg 21004
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 12934 . . 3 (𝑧 ∈ β„š ↔ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„• 𝑧 = (π‘₯ / 𝑦))
2 drngring 20364 . . . . . . . 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 21003 . . . . . . . . . 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 2733 . . . . . . . . . . 11 (β„‚fld β†Ύs 𝑅) = (β„‚fld β†Ύs 𝑅)
76subrgbas 20328 . . . . . . . . . 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 4023 . . . . . . . 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 3984 . . . . . . 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 12579 . . . . . . . . . 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 3984 . . . . . . . 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 12246 . . . . . . . . . 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 20969 . . . . . . . . . . 11 0 = (0gβ€˜β„‚fld)
186, 17subrg0 20326 . . . . . . . . . 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 3011 . . . . . . . 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 2733 . . . . . . . . . 10 (Baseβ€˜(β„‚fld β†Ύs 𝑅)) = (Baseβ€˜(β„‚fld β†Ύs 𝑅))
22 eqid 2733 . . . . . . . . . 10 (Unitβ€˜(β„‚fld β†Ύs 𝑅)) = (Unitβ€˜(β„‚fld β†Ύs 𝑅))
23 eqid 2733 . . . . . . . . . 10 (0gβ€˜(β„‚fld β†Ύs 𝑅)) = (0gβ€˜(β„‚fld β†Ύs 𝑅))
2421, 22, 23drngunit 20362 . . . . . . . . 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 2733 . . . . . . . 8 (/rβ€˜(β„‚fld β†Ύs 𝑅)) = (/rβ€˜(β„‚fld β†Ύs 𝑅))
2821, 22, 27dvrcl 20218 . . . . . . 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 3984 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ π‘₯ ∈ 𝑅)
32 cnflddiv 20975 . . . . . . . 8 / = (/rβ€˜β„‚fld)
336, 32, 22, 27subrgdv 20336 . . . . . . 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 2849 . . . . 5 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (π‘₯ / 𝑦) ∈ 𝑅)
36 eleq1 2822 . . . . 5 (𝑧 = (π‘₯ / 𝑦) β†’ (𝑧 ∈ 𝑅 ↔ (π‘₯ / 𝑦) ∈ 𝑅))
3735, 36syl5ibrcom 246 . . . 4 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (𝑧 = (π‘₯ / 𝑦) β†’ 𝑧 ∈ 𝑅))
3837rexlimdvva 3212 . . 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 3989 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 2941  βˆƒwrex 3071   βŠ† wss 3949  β€˜cfv 6544  (class class class)co 7409  0cc0 11110   / cdiv 11871  β„•cn 12212  β„€cz 12558  β„šcq 12932  Basecbs 17144   β†Ύs cress 17173  0gc0g 17385  Ringcrg 20056  Unitcui 20169  /rcdvr 20214  SubRingcsubrg 20315  DivRingcdr 20357  β„‚fldccnfld 20944
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-addf 11189  ax-mulf 11190
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-fz 13485  df-seq 13967  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-mulg 18951  df-subg 19003  df-cmn 19650  df-mgp 19988  df-ur 20005  df-ring 20058  df-cring 20059  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-invr 20202  df-dvr 20215  df-subrg 20317  df-drng 20359  df-cnfld 20945
This theorem is referenced by:  cphqss  24705  resscdrg  24875
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