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Theorem qsssubdrg 21003
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 12933 . . 3 (𝑧 ∈ β„š ↔ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„• 𝑧 = (π‘₯ / 𝑦))
2 drngring 20363 . . . . . . . 8 ((β„‚fld β†Ύs 𝑅) ∈ DivRing β†’ (β„‚fld β†Ύs 𝑅) ∈ Ring)
32ad2antlr 725 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (β„‚fld β†Ύs 𝑅) ∈ Ring)
4 zsssubrg 21002 . . . . . . . . . 10 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ β„€ βŠ† 𝑅)
54ad2antrr 724 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ β„€ βŠ† 𝑅)
6 eqid 2732 . . . . . . . . . . 11 (β„‚fld β†Ύs 𝑅) = (β„‚fld β†Ύs 𝑅)
76subrgbas 20327 . . . . . . . . . 10 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ 𝑅 = (Baseβ€˜(β„‚fld β†Ύs 𝑅)))
87ad2antrr 724 . . . . . . . . 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 4022 . . . . . . . 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 769 . . . . . . . 8 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ π‘₯ ∈ β„€)
119, 10sseldd 3983 . . . . . . 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 12578 . . . . . . . . . 10 (𝑦 ∈ β„• β†’ 𝑦 ∈ β„€)
1312ad2antll 727 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 ∈ β„€)
149, 13sseldd 3983 . . . . . . . 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 12245 . . . . . . . . . 10 (𝑦 ∈ β„• β†’ 𝑦 β‰  0)
1615ad2antll 727 . . . . . . . . 9 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑦 β‰  0)
17 cnfld0 20968 . . . . . . . . . . 11 0 = (0gβ€˜β„‚fld)
186, 17subrg0 20325 . . . . . . . . . 10 (𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ 0 = (0gβ€˜(β„‚fld β†Ύs 𝑅)))
1918ad2antrr 724 . . . . . . . . 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 3010 . . . . . . . 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 2732 . . . . . . . . . 10 (Baseβ€˜(β„‚fld β†Ύs 𝑅)) = (Baseβ€˜(β„‚fld β†Ύs 𝑅))
22 eqid 2732 . . . . . . . . . 10 (Unitβ€˜(β„‚fld β†Ύs 𝑅)) = (Unitβ€˜(β„‚fld β†Ύs 𝑅))
23 eqid 2732 . . . . . . . . . 10 (0gβ€˜(β„‚fld β†Ύs 𝑅)) = (0gβ€˜(β„‚fld β†Ύs 𝑅))
2421, 22, 23drngunit 20361 . . . . . . . . 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 725 . . . . . . . 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 711 . . . . . . 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 2732 . . . . . . . 8 (/rβ€˜(β„‚fld β†Ύs 𝑅)) = (/rβ€˜(β„‚fld β†Ύs 𝑅))
2821, 22, 27dvrcl 20217 . . . . . . 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 1371 . . . . . 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 765 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ 𝑅 ∈ (SubRingβ€˜β„‚fld))
315, 10sseldd 3983 . . . . . . 7 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ π‘₯ ∈ 𝑅)
32 cnflddiv 20974 . . . . . . . 8 / = (/rβ€˜β„‚fld)
336, 32, 22, 27subrgdv 20335 . . . . . . 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 1371 . . . . . 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 2848 . . . . 5 (((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„•)) β†’ (π‘₯ / 𝑦) ∈ 𝑅)
36 eleq1 2821 . . . . 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 3211 . . 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 3988 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   βŠ† wss 3948  β€˜cfv 6543  (class class class)co 7408  0cc0 11109   / cdiv 11870  β„•cn 12211  β„€cz 12557  β„šcq 12931  Basecbs 17143   β†Ύs cress 17172  0gc0g 17384  Ringcrg 20055  Unitcui 20168  /rcdvr 20213  SubRingcsubrg 20314  DivRingcdr 20356  β„‚fldccnfld 20943
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-addf 11188  ax-mulf 11189
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-tpos 8210  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  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 12472  df-z 12558  df-dec 12677  df-uz 12822  df-q 12932  df-fz 13484  df-seq 13966  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-starv 17211  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-mulg 18950  df-subg 19002  df-cmn 19649  df-mgp 19987  df-ur 20004  df-ring 20057  df-cring 20058  df-oppr 20149  df-dvdsr 20170  df-unit 20171  df-invr 20201  df-dvr 20214  df-subrg 20316  df-drng 20358  df-cnfld 20944
This theorem is referenced by:  cphqss  24704  resscdrg  24874
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