Theorem eleesub 23304

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 23294	. . . . . 6 |- ( ( A e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. RR )
3		fveere 23294	. . . . . 6 |- ( ( B e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( B ` i ) e. RR )
4		resubcl 9779	. . . . . 6 |- ( ( ( A ` i ) e. RR /\ ( B ` i ) e. RR ) -> ( ( A ` i ) - ( B ` i ) ) e. RR )
5	2, 3, 4	syl2an 477	. . . . 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 827	. . . 4 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ i e. ( 1 ... N ) ) -> ( ( A ` i ) - ( B ` i ) ) e. RR )
7	6	ralrimiva 2827	. . 3 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> A. i e. ( 1 ... N ) ( ( A ` i ) - ( B ` i ) ) e. RR )
8		eleenn 23289	. . . . 5 |- ( A e. ( EE ` N ) -> N e. NN )
9		mptelee 23288	. . . . 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 465	. . 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 2544	1 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> C e. ( EE ` N ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2796   |-> cmpt 4453   ` cfv 5521  (class class class)co 6195   RRcr 9387   1c1 9389   - cmin 9701   NNcn 10428   ...cfz 11549   EEcee 23281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-ltxr 9529  df-sub 9703  df-neg 9704  df-ee 23284

This theorem is referenced by: (None)