Theorem eleesub 24416

Description: Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.)

Hypothesis

Ref	Expression
eleesub.1	|- C = ( i e. ( 1 ... N ) |-> ( ( A ` i ) - ( B ` i ) ) )

Assertion

Ref	Expression
eleesub	|- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> C e. ( EE ` N ) )

Distinct variable groups:   i, N   A, i   B, i

Allowed substitution hint:   C( i)

Proof of Theorem eleesub

Step	Hyp	Ref	Expression
1		eleesub.1	. 2 |- C = ( i e. ( 1 ... N ) |-> ( ( A ` i ) - ( B ` i ) ) )
2		fveere 24406	. . . . . 6 |- ( ( A e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. RR )
3		fveere 24406	. . . . . 6 |- ( ( B e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( B ` i ) e. RR )
4		resubcl 9874	. . . . . 6 |- ( ( ( A ` i ) e. RR /\ ( B ` i ) e. RR ) -> ( ( A ` i ) - ( B ` i ) ) e. RR )
5	2, 3, 4	syl2an 475	. . . . 5 |- ( ( ( A e. ( EE ` N ) /\ i e. ( 1 ... N ) ) /\ ( B e. ( EE ` N ) /\ i e. ( 1 ... N ) ) ) -> ( ( A ` i ) - ( B ` i ) ) e. RR )
6	5	anandirs 829	. . . 4 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ i e. ( 1 ... N ) ) -> ( ( A ` i ) - ( B ` i ) ) e. RR )
7	6	ralrimiva 2868	. . 3 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> A. i e. ( 1 ... N ) ( ( A ` i ) - ( B ` i ) ) e. RR )
8		eleenn 24401	. . . . 5 |- ( A e. ( EE ` N ) -> N e. NN )
9		mptelee 24400	. . . . 5 |- ( N e. NN -> ( ( i e. ( 1 ... N ) |-> ( ( A ` i ) - ( B ` i ) ) ) e. ( EE ` N ) <-> A. i e. ( 1 ... N ) ( ( A ` i ) - ( B ` i ) ) e. RR ) )
10	8, 9	syl 16	. . . 4 |- ( A e. ( EE ` N ) -> ( ( i e. ( 1 ... N ) |-> ( ( A ` i ) - ( B ` i ) ) ) e. ( EE ` N ) <-> A. i e. ( 1 ... N ) ( ( A ` i ) - ( B ` i ) ) e. RR ) )
11	10	adantr 463	. . 3 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( ( i e. ( 1 ... N ) |-> ( ( A ` i ) - ( B ` i ) ) ) e. ( EE ` N ) <-> A. i e. ( 1 ... N ) ( ( A ` i ) - ( B ` i ) ) e. RR ) )
12	7, 11	mpbird 232	. 2 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( i e. ( 1 ... N ) |-> ( ( A ` i ) - ( B ` i ) ) ) e. ( EE ` N ) )
13	1, 12	syl5eqel 2546	1 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> C e. ( EE ` N ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   A.wral 2804   |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   RRcr 9480   1c1 9482   - cmin 9796   NNcn 10531   ...cfz 11675   EEcee 24393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-neg 9799  df-ee 24396

This theorem is referenced by: (None)