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Theorem modqid 10799
Description: Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
Assertion
Ref Expression
modqid |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
Proof of Theorem modqid
StepHypRef Expression
1 simpll 531 . . 3 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> A e. QQ )
2 simplr 533 . . 3 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> B e. QQ )
3 0red 8327 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> 0 e. RR )
4 qre 10034 . . . . 5 |- ( A e. QQ -> A e. RR )
54ad2antrr 492 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> A e. RR )
6 qre 10034 . . . . 5 |- ( B e. QQ -> B e. RR )
76ad2antlr 493 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> B e. RR )
8 simprl 535 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> 0 <_ A )
9 simprr 537 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> A < B )
103, 5, 7, 8, 9lelttrd 8452 . . 3 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> 0 < B )
11 modqval 10774 . . 3 |- ( ( A e. QQ /\ B e. QQ /\ 0 < B ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
121, 2, 10, 11syl3anc 1278 . 2 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
1310gt0ne0d 8841 . . . . . . . . 9 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> B =/= 0 )
14 qdivcl 10052 . . . . . . . . 9 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
151, 2, 13, 14syl3anc 1278 . . . . . . . 8 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A / B ) e. QQ )
16 qcn 10043 . . . . . . . 8 |- ( ( A / B ) e. QQ -> ( A / B ) e. CC )
17 addlid 8466 . . . . . . . . 9 |- ( ( A / B ) e. CC -> ( 0 + ( A / B ) ) = ( A / B ) )
1817fveq2d 5699 . . . . . . . 8 |- ( ( A / B ) e. CC -> ( |_ ` ( 0 + ( A / B ) ) ) = ( |_ ` ( A / B ) ) )
1915, 16, 183syl 17 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( |_ ` ( 0 + ( A / B ) ) ) = ( |_ ` ( A / B ) ) )
20 divge0 9205 . . . . . . . . 9 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) )
215, 8, 7, 10, 20syl22anc 1279 . . . . . . . 8 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> 0 <_ ( A / B ) )
227recnd 8354 . . . . . . . . . . 11 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> B e. CC )
2322mulridd 8343 . . . . . . . . . 10 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( B x. 1 ) = B )
249, 23breqtrrd 4158 . . . . . . . . 9 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> A < ( B x. 1 ) )
25 1red 8341 . . . . . . . . . 10 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> 1 e. RR )
26 ltdivmul 9208 . . . . . . . . . 10 |- ( ( A e. RR /\ 1 e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) < 1 <-> A < ( B x. 1 ) ) )
275, 25, 7, 10, 26syl112anc 1282 . . . . . . . . 9 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( ( A / B ) < 1 <-> A < ( B x. 1 ) ) )
2824, 27mpbird 167 . . . . . . . 8 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A / B ) < 1 )
29 0z 9659 . . . . . . . . 9 |- 0 e. ZZ
30 flqbi2 10739 . . . . . . . . 9 |- ( ( 0 e. ZZ /\ ( A / B ) e. QQ ) -> ( ( |_ ` ( 0 + ( A / B ) ) ) = 0 <-> ( 0 <_ ( A / B ) /\ ( A / B ) < 1 ) ) )
3129, 15, 30sylancr 418 . . . . . . . 8 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( ( |_ ` ( 0 + ( A / B ) ) ) = 0 <-> ( 0 <_ ( A / B ) /\ ( A / B ) < 1 ) ) )
3221, 28, 31mpbir2and 957 . . . . . . 7 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( |_ ` ( 0 + ( A / B ) ) ) = 0 )
3319, 32eqtr3d 2273 . . . . . 6 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( |_ ` ( A / B ) ) = 0 )
3433oveq2d 6101 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( B x. ( |_ ` ( A / B ) ) ) = ( B x. 0 ) )
3522mul01d 8721 . . . . 5 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( B x. 0 ) = 0 )
3634, 35eqtrd 2271 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( B x. ( |_ ` ( A / B ) ) ) = 0 )
3736oveq2d 6101 . . 3 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) = ( A - 0 ) )
385recnd 8354 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> A e. CC )
3938subid1d 8627 . . 3 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A - 0 ) = A )
4037, 39eqtrd 2271 . 2 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A - ( B x. ( |_ ` ( A / B ) ) ) ) = A )
4112, 40eqtrd 2271 1 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> ( A mod B ) = A )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   - cmin 8498   / cdiv 9004   ZZcz 9648   QQcq 10028   |_cfl 10713   mod cmo 10772
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-n0 9568  df-z 9649  df-q 10029  df-rp 10065  df-fl 10715  df-mod 10773
This theorem is used by:  modqid2  10801  q0mod  10805  q1mod  10806  modqabs  10807  mulqaddmodid  10814  m1modnnsub1  10820  modqltm1p1mod  10826  q2submod  10835  modifeq2int  10836  modaddmodlo  10838  modqsubdir  10843  modsumfzodifsn  10846  bitsinv1  12745  crth  13022  eulerthlemh  13029  prmdiveq  13034  modprm0  13053  4sqlem12  13201  znf1o  15035  wilthlem1  16151  ppiqub  16194  lgslem1  16217  lgsdir2lem1  16245  lgsdirprm  16251  lgseisenlem1  16287  lgseisenlem2  16288  lgseisen  16291  m1lgs  16302  2lgslem1a1  16303  2lgslem4  16320
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