Theorem addpipq2 9758

Description: Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
addpipq2	|- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A +pQ B ) = <. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )

Proof of Theorem addpipq2

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( x = A -> ( 1st ` x ) = ( 1st ` A ) )
2	1	oveq1d 6665	. . . 4 |- ( x = A -> ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` A ) .N ( 2nd ` y ) ) )
3		fveq2 6191	. . . . 5 |- ( x = A -> ( 2nd ` x ) = ( 2nd ` A ) )
4	3	oveq2d 6666	. . . 4 |- ( x = A -> ( ( 1st ` y ) .N ( 2nd ` x ) ) = ( ( 1st ` y ) .N ( 2nd ` A ) ) )
5	2, 4	oveq12d 6668	. . 3 |- ( x = A -> ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) = ( ( ( 1st ` A ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` A ) ) ) )
6	3	oveq1d 6665	. . 3 |- ( x = A -> ( ( 2nd ` x ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` y ) ) )
7	5, 6	opeq12d 4410	. 2 |- ( x = A -> <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. = <. ( ( ( 1st ` A ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` y ) ) >. )
8		fveq2 6191	. . . . 5 |- ( y = B -> ( 2nd ` y ) = ( 2nd ` B ) )
9	8	oveq2d 6666	. . . 4 |- ( y = B -> ( ( 1st ` A ) .N ( 2nd ` y ) ) = ( ( 1st ` A ) .N ( 2nd ` B ) ) )
10		fveq2 6191	. . . . 5 |- ( y = B -> ( 1st ` y ) = ( 1st ` B ) )
11	10	oveq1d 6665	. . . 4 |- ( y = B -> ( ( 1st ` y ) .N ( 2nd ` A ) ) = ( ( 1st ` B ) .N ( 2nd ` A ) ) )
12	9, 11	oveq12d 6668	. . 3 |- ( y = B -> ( ( ( 1st ` A ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` A ) ) ) = ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
13	8	oveq2d 6666	. . 3 |- ( y = B -> ( ( 2nd ` A ) .N ( 2nd ` y ) ) = ( ( 2nd ` A ) .N ( 2nd ` B ) ) )
14	12, 13	opeq12d 4410	. 2 |- ( y = B -> <. ( ( ( 1st ` A ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` y ) ) >. = <. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )
15		df-plpq 9730	. 2 |- +pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) +N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
16		opex 4932	. 2 |- <. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. e. _V
17	7, 14, 15, 16	ovmpt2 6796	1 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A +pQ B ) = <. ( ( ( 1st ` A ) .N ( 2nd ` B ) ) +N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) , ( ( 2nd ` A ) .N ( 2nd ` B ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   +N cpli 9667   .N cmi 9668   +pQ cplpq 9670

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-plpq 9730

This theorem is referenced by:  addpipq  9759  addcompq  9772  adderpqlem  9776  addassnq  9780  distrnq  9783  ltanq  9793