Theorem dvhopaddN 36403

Description: Sum of DVecH vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dvhopadd.a	|- A = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) P ( 2nd ` g ) ) >. )

Assertion

Ref	Expression
dvhopaddN	|- ( ( ( F e. T /\ U e. E ) /\ ( G e. T /\ V e. E ) ) -> ( <. F , U >. A <. G , V >. ) = <. ( F o. G ) , ( U P V ) >. )

Distinct variable groups:   f, g, E   P, f, g   T, f, g

Allowed substitution hints:   A( f, g)   U( f, g)   F( f, g)   G( f, g)   V( f, g)

Proof of Theorem dvhopaddN

Step	Hyp	Ref	Expression
1		opelxpi 5148	. . 3 |- ( ( F e. T /\ U e. E ) -> <. F , U >. e. ( T X. E ) )
2		opelxpi 5148	. . 3 |- ( ( G e. T /\ V e. E ) -> <. G , V >. e. ( T X. E ) )
3		dvhopadd.a	. . . 4 |- A = ( f e. ( T X. E ) , g e. ( T X. E ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) P ( 2nd ` g ) ) >. )
4	3	dvhvaddval 36379	. . 3 |- ( ( <. F , U >. e. ( T X. E ) /\ <. G , V >. e. ( T X. E ) ) -> ( <. F , U >. A <. G , V >. ) = <. ( ( 1st ` <. F , U >. ) o. ( 1st ` <. G , V >. ) ) , ( ( 2nd ` <. F , U >. ) P ( 2nd ` <. G , V >. ) ) >. )
5	1, 2, 4	syl2an 494	. 2 |- ( ( ( F e. T /\ U e. E ) /\ ( G e. T /\ V e. E ) ) -> ( <. F , U >. A <. G , V >. ) = <. ( ( 1st ` <. F , U >. ) o. ( 1st ` <. G , V >. ) ) , ( ( 2nd ` <. F , U >. ) P ( 2nd ` <. G , V >. ) ) >. )
6		op1stg 7180	. . . . 5 |- ( ( F e. T /\ U e. E ) -> ( 1st ` <. F , U >. ) = F )
7	6	adantr 481	. . . 4 |- ( ( ( F e. T /\ U e. E ) /\ ( G e. T /\ V e. E ) ) -> ( 1st ` <. F , U >. ) = F )
8		op1stg 7180	. . . . 5 |- ( ( G e. T /\ V e. E ) -> ( 1st ` <. G , V >. ) = G )
9	8	adantl 482	. . . 4 |- ( ( ( F e. T /\ U e. E ) /\ ( G e. T /\ V e. E ) ) -> ( 1st ` <. G , V >. ) = G )
10	7, 9	coeq12d 5286	. . 3 |- ( ( ( F e. T /\ U e. E ) /\ ( G e. T /\ V e. E ) ) -> ( ( 1st ` <. F , U >. ) o. ( 1st ` <. G , V >. ) ) = ( F o. G ) )
11		op2ndg 7181	. . . 4 |- ( ( F e. T /\ U e. E ) -> ( 2nd ` <. F , U >. ) = U )
12		op2ndg 7181	. . . 4 |- ( ( G e. T /\ V e. E ) -> ( 2nd ` <. G , V >. ) = V )
13	11, 12	oveqan12d 6669	. . 3 |- ( ( ( F e. T /\ U e. E ) /\ ( G e. T /\ V e. E ) ) -> ( ( 2nd ` <. F , U >. ) P ( 2nd ` <. G , V >. ) ) = ( U P V ) )
14	10, 13	opeq12d 4410	. 2 |- ( ( ( F e. T /\ U e. E ) /\ ( G e. T /\ V e. E ) ) -> <. ( ( 1st ` <. F , U >. ) o. ( 1st ` <. G , V >. ) ) , ( ( 2nd ` <. F , U >. ) P ( 2nd ` <. G , V >. ) ) >. = <. ( F o. G ) , ( U P V ) >. )
15	5, 14	eqtrd 2656	1 |- ( ( ( F e. T /\ U e. E ) /\ ( G e. T /\ V e. E ) ) -> ( <. F , U >. A <. G , V >. ) = <. ( F o. G ) , ( U P V ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  dvhopN  36405