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Theorem quotcan 23130
 Description: Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
quotcan.1
Assertion
Ref Expression
quotcan Poly Poly quot

Proof of Theorem quotcan
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 23022 . . . . . . . . 9 Poly Poly
2 simp2 1006 . . . . . . . . 9 Poly Poly Poly
31, 2sseldi 3468 . . . . . . . 8 Poly Poly Poly
4 simp1 1005 . . . . . . . . . 10 Poly Poly Poly
51, 4sseldi 3468 . . . . . . . . 9 Poly Poly Poly
6 quotcan.1 . . . . . . . . . . . 12
7 plymulcl 23043 . . . . . . . . . . . 12 Poly Poly Poly
86, 7syl5eqel 2521 . . . . . . . . . . 11 Poly Poly Poly
983adant3 1025 . . . . . . . . . 10 Poly Poly Poly
10 simp3 1007 . . . . . . . . . 10 Poly Poly
11 quotcl2 23123 . . . . . . . . . 10 Poly Poly quot Poly
129, 3, 10, 11syl3anc 1264 . . . . . . . . 9 Poly Poly quot Poly
13 plysubcl 23044 . . . . . . . . 9 Poly quot Poly quot Poly
145, 12, 13syl2anc 665 . . . . . . . 8 Poly Poly quot Poly
15 plymul0or 23102 . . . . . . . 8 Poly quot Poly quot quot
163, 14, 15syl2anc 665 . . . . . . 7 Poly Poly quot quot
17 cnex 9619 . . . . . . . . . . . . 13
1817a1i 11 . . . . . . . . . . . 12 Poly Poly
19 plyf 23020 . . . . . . . . . . . . 13 Poly
204, 19syl 17 . . . . . . . . . . . 12 Poly Poly
21 plyf 23020 . . . . . . . . . . . . 13 Poly
222, 21syl 17 . . . . . . . . . . . 12 Poly Poly
23 mulcom 9624 . . . . . . . . . . . . 13
2423adantl 467 . . . . . . . . . . . 12 Poly Poly
2518, 20, 22, 24caofcom 6577 . . . . . . . . . . 11 Poly Poly
266, 25syl5eq 2482 . . . . . . . . . 10 Poly Poly
2726oveq1d 6320 . . . . . . . . 9 Poly Poly quot quot
28 plyf 23020 . . . . . . . . . . 11 quot Poly quot
2912, 28syl 17 . . . . . . . . . 10 Poly Poly quot
30 subdi 10051 . . . . . . . . . . 11
3130adantl 467 . . . . . . . . . 10 Poly Poly
3218, 22, 20, 29, 31caofdi 6581 . . . . . . . . 9 Poly Poly quot quot
3327, 32eqtr4d 2473 . . . . . . . 8 Poly Poly quot quot
3433eqeq1d 2431 . . . . . . 7 Poly Poly quot quot
3510neneqd 2632 . . . . . . . 8 Poly Poly
36 biorf 406 . . . . . . . 8 quot quot
3735, 36syl 17 . . . . . . 7 Poly Poly quot quot
3816, 34, 373bitr4d 288 . . . . . 6 Poly Poly quot quot
3938biimpd 210 . . . . 5 Poly Poly quot quot
40 eqid 2429 . . . . . . . . . . 11 deg deg
41 eqid 2429 . . . . . . . . . . 11 deg quot deg quot
4240, 41dgrmul 23092 . . . . . . . . . 10 Poly quot Poly quot deg quot deg deg quot
4342expr 618 . . . . . . . . 9 Poly quot Poly quot deg quot deg deg quot
443, 10, 14, 43syl21anc 1263 . . . . . . . 8 Poly Poly quot deg quot deg deg quot
45 dgrcl 23055 . . . . . . . . . . . 12 Poly deg
462, 45syl 17 . . . . . . . . . . 11 Poly Poly deg
4746nn0red 10926 . . . . . . . . . 10 Poly Poly deg
48 dgrcl 23055 . . . . . . . . . . 11 quot Poly deg quot
4914, 48syl 17 . . . . . . . . . 10 Poly Poly deg quot
50 nn0addge1 10916 . . . . . . . . . 10 deg deg quot deg deg deg quot
5147, 49, 50syl2anc 665 . . . . . . . . 9 Poly Poly deg deg deg quot
52 breq2 4430 . . . . . . . . 9 deg quot deg deg quot deg deg quot deg deg deg quot
5351, 52syl5ibrcom 225 . . . . . . . 8 Poly Poly deg quot deg deg quot deg deg quot
5444, 53syld 45 . . . . . . 7 Poly Poly quot deg deg quot
5533fveq2d 5885 . . . . . . . . 9 Poly Poly deg quot deg quot
5655breq2d 4438 . . . . . . . 8 Poly Poly deg deg quot deg deg quot
57 plymulcl 23043 . . . . . . . . . . . . 13 Poly quot Poly quot Poly
583, 12, 57syl2anc 665 . . . . . . . . . . . 12 Poly Poly quot Poly
59 plysubcl 23044 . . . . . . . . . . . 12 Poly quot Poly quot Poly
609, 58, 59syl2anc 665 . . . . . . . . . . 11 Poly Poly quot Poly
61 dgrcl 23055 . . . . . . . . . . 11 quot Poly deg quot
6260, 61syl 17 . . . . . . . . . 10 Poly Poly deg quot
6362nn0red 10926 . . . . . . . . 9 Poly Poly deg quot
6447, 63lenltd 9780 . . . . . . . 8 Poly Poly deg deg quot deg quot deg
6556, 64bitr3d 258 . . . . . . 7 Poly Poly deg deg quot deg quot deg
6654, 65sylibd 217 . . . . . 6 Poly Poly quot deg quot deg
6766necon4ad 2651 . . . . 5 Poly Poly deg quot deg quot
68 eqid 2429 . . . . . . 7 quot quot
6968quotdgr 23124 . . . . . 6 Poly Poly quot deg quot deg
709, 3, 10, 69syl3anc 1264 . . . . 5 Poly Poly quot deg quot deg
7139, 67, 70mpjaod 382 . . . 4 Poly Poly quot
72 df-0p 22505 . . . 4
7371, 72syl6eq 2486 . . 3 Poly Poly quot
74 ofsubeq0 10606 . . . 4 quot quot quot
7518, 20, 29, 74syl3anc 1264 . . 3 Poly Poly quot quot
7673, 75mpbid 213 . 2 Poly Poly quot
7776eqcomd 2437 1 Poly Poly quot
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wo 369   wa 370   w3a 982   wceq 1437   wcel 1870   wne 2625  cvv 3087  csn 4002   class class class wbr 4426   cxp 4852  wf 5597  cfv 5601  (class class class)co 6305   cof 6543  cc 9536  cr 9537  cc0 9538   caddc 9541   cmul 9543   clt 9674   cle 9675   cmin 9859  cn0 10869  c0p 22504  Polycply 23006  degcdgr 23009   quot cquot 23111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-rlim 13531  df-sum 13731  df-0p 22505  df-ply 23010  df-coe 23012  df-dgr 23013  df-quot 23112 This theorem is referenced by: (None)
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