Theorem modval 12670

Description: The value of the modulo operation. The modulo congruence notation of number theory, J == K (modulo N), can be expressed in our notation as ( J mod N ) = ( K mod N ). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive reals to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) (Contributed by NM, 10-Nov-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)

Assertion

Ref	Expression
modval	|- ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )

Proof of Theorem modval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . 5 |- ( x = A -> ( x / y ) = ( A / y ) )
2	1	fveq2d 6195	. . . 4 |- ( x = A -> ( |_ ` ( x / y ) ) = ( |_ ` ( A / y ) ) )
3	2	oveq2d 6666	. . 3 |- ( x = A -> ( y x. ( |_ ` ( x / y ) ) ) = ( y x. ( |_ ` ( A / y ) ) ) )
4		oveq12 6659	. . 3 |- ( ( x = A /\ ( y x. ( |_ ` ( x / y ) ) ) = ( y x. ( |_ ` ( A / y ) ) ) ) -> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) = ( A - ( y x. ( |_ ` ( A / y ) ) ) ) )
5	3, 4	mpdan 702	. 2 |- ( x = A -> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) = ( A - ( y x. ( |_ ` ( A / y ) ) ) ) )
6		oveq2 6658	. . . . 5 |- ( y = B -> ( A / y ) = ( A / B ) )
7	6	fveq2d 6195	. . . 4 |- ( y = B -> ( |_ ` ( A / y ) ) = ( |_ ` ( A / B ) ) )
8		oveq12 6659	. . . 4 |- ( ( y = B /\ ( |_ ` ( A / y ) ) = ( |_ ` ( A / B ) ) ) -> ( y x. ( |_ ` ( A / y ) ) ) = ( B x. ( |_ ` ( A / B ) ) ) )
9	7, 8	mpdan 702	. . 3 |- ( y = B -> ( y x. ( |_ ` ( A / y ) ) ) = ( B x. ( |_ ` ( A / B ) ) ) )
10	9	oveq2d 6666	. 2 |- ( y = B -> ( A - ( y x. ( |_ ` ( A / y ) ) ) ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
11		df-mod 12669	. 2 |- mod = ( x e. RR , y e. RR+ |-> ( x - ( y x. ( |_ ` ( x / y ) ) ) ) )
12		ovex 6678	. 2 |- ( A - ( B x. ( |_ ` ( A / B ) ) ) ) e. _V
13	5, 10, 11, 12	ovmpt2 6796	1 |- ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   RRcr 9935   x. cmul 9941   - cmin 10266   / cdiv 10684   RR+crp 11832   |_cfl 12591   mod cmo 12668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mod 12669

This theorem is referenced by:  modvalr  12671  modcl  12672  mod0  12675  modge0  12678  modlt  12679  moddiffl  12681  modfrac  12683  modmulnn  12688  zmodcl  12690  modid  12695  modcyc  12705  modadd1  12707  modmul1  12723  moddi  12738  modsubdir  12739  modirr  12741  iexpcyc  12969  digit2  12997  dvdsmod  15050  divalgmod  15129  divalgmodOLD  15130  modgcd  15253  bezoutlem3  15258  prmdiv  15490  odzdvds  15500  fldivp1  15601  mulgmodid  17581  odmodnn0  17959  odmod  17965  gexdvds  17999  zringlpirlem3  19834  sineq0  24273  efif1olem2  24289  lgseisenlem4  25103  dchrisumlem1  25178  ostth2lem2  25323  sineq0ALT  39173  ltmod  39870  fourierswlem  40447