Theorem climadd 11637

Description: Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)

Hypotheses

Ref	Expression
climadd.1	|- Z = ( ZZ>= ` M )
climadd.2	|- ( ph -> M e. ZZ )
climadd.4	|- ( ph -> F ~~> A )
climadd.6	|- ( ph -> H e. X )
climadd.7	|- ( ph -> G ~~> B )
climadd.8	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
climadd.9	|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
climadd.h	|- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) + ( G ` k ) ) )

Assertion

Ref	Expression
climadd	|- ( ph -> H ~~> ( A + B ) )

Distinct variable groups:   B, k   k, F   ph, k   A, k   k, G   k, H   k, M   k, Z

Allowed substitution hint:   X( k)

Proof of Theorem climadd

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

Step	Hyp	Ref	Expression
1		climadd.1	. 2 |- Z = ( ZZ>= ` M )
2		climadd.2	. 2 |- ( ph -> M e. ZZ )
3		climadd.4	. . 3 |- ( ph -> F ~~> A )
4		climcl 11593	. . 3 |- ( F ~~> A -> A e. CC )
5	3, 4	syl 14	. 2 |- ( ph -> A e. CC )
6		climadd.7	. . 3 |- ( ph -> G ~~> B )
7		climcl 11593	. . 3 |- ( G ~~> B -> B e. CC )
8	6, 7	syl 14	. 2 |- ( ph -> B e. CC )
9		addcl 8050	. . 3 |- ( ( u e. CC /\ v e. CC ) -> ( u + v ) e. CC )
10	9	adantl 277	. 2 |- ( ( ph /\ ( u e. CC /\ v e. CC ) ) -> ( u + v ) e. CC )
11		climadd.6	. 2 |- ( ph -> H e. X )
12		simpr 110	. . 3 |- ( ( ph /\ x e. RR+ ) -> x e. RR+ )
13	5	adantr 276	. . 3 |- ( ( ph /\ x e. RR+ ) -> A e. CC )
14	8	adantr 276	. . 3 |- ( ( ph /\ x e. RR+ ) -> B e. CC )
15		addcn2 11621	. . 3 |- ( ( x e. RR+ /\ A e. CC /\ B e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - A ) ) < y /\ ( abs ` ( v - B ) ) < z ) -> ( abs ` ( ( u + v ) - ( A + B ) ) ) < x ) )
16	12, 13, 14, 15	syl3anc 1250	. 2 |- ( ( ph /\ x e. RR+ ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - A ) ) < y /\ ( abs ` ( v - B ) ) < z ) -> ( abs ` ( ( u + v ) - ( A + B ) ) ) < x ) )
17		climadd.8	. 2 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
18		climadd.9	. 2 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
19		climadd.h	. 2 |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) + ( G ` k ) ) )
20	1, 2, 5, 8, 10, 3, 6, 11, 16, 17, 18, 19	climcn2 11620	1 |- ( ph -> H ~~> ( A + B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   E.wrex 2485   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   CCcc 7923   + caddc 7928   < clt 8107   - cmin 8243   ZZcz 9372   ZZ>=cuz 9648   RR+crp 9775   abscabs 11308   ~~> cli 11589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-rp 9776  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590

This theorem is referenced by:  climaddc1  11640  climcvg1nlem  11660  isumadd  11742