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Theorem modqid 10801
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 8328 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> 0 e. RR )
4 qre 10035 . . . . 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 10035 . . . . 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 8453 . . 3 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> 0 < B )
11 modqval 10776 . . 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 8842 . . . . . . . . 9 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> B =/= 0 )
14 qdivcl 10053 . . . . . . . . 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 10044 . . . . . . . 8 |- ( ( A / B ) e. QQ -> ( A / B ) e. CC )
17 addlid 8467 . . . . . . . . 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 9206 . . . . . . . . 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 8355 . . . . . . . . . . 11 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> B e. CC )
2322mulridd 8344 . . . . . . . . . 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 8342 . . . . . . . . . 10 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> 1 e. RR )
26 ltdivmul 9209 . . . . . . . . . 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 9660 . . . . . . . . 9 |- 0 e. ZZ
30 flqbi2 10741 . . . . . . . . 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 8722 . . . . 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 8355 . . . 4 |- ( ( ( A e. QQ /\ B e. QQ ) /\ ( 0 <_ A /\ A < B ) ) -> A e. CC )
3938subid1d 8628 . . 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 8178   RRcr 8179   0cc0 8180   1c1 8181   + caddc 8183   x. cmul 8185   < clt 8361   <_ cle 8362   - cmin 8499   / cdiv 9005   ZZcz 9649   QQcq 10029   |_cfl 10714   mod cmo 10774
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 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299
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 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-n0 9569  df-z 9650  df-q 10030  df-rp 10066  df-fl 10716  df-mod 10775
This theorem is used by:  modqid2  10803  q0mod  10807  q1mod  10808  modqabs  10809  mulqaddmodid  10816  m1modnnsub1  10822  modqltm1p1mod  10828  q2submod  10837  modifeq2int  10838  modaddmodlo  10840  modqsubdir  10845  modsumfzodifsn  10848  bitsinv1  12748  crth  13025  eulerthlemh  13032  prmdiveq  13037  modprm0  13056  4sqlem12  13204  znf1o  15070  wilthlem1  16193  ppiqub  16254  lgslem1  16285  lgsdir2lem1  16313  lgsdirprm  16319  lgseisenlem1  16355  lgseisenlem2  16356  lgseisen  16359  m1lgs  16370  2lgslem1a1  16371  2lgslem4  16388
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