Theorem climadd 11966

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 11922	. . 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 11922	. . 3 |- ( G ~~> B -> B e. CC )
8	6, 7	syl 14	. 2 |- ( ph -> B e. CC )
9		addcl 8217	. . 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 11950	. . 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 1274	. 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 11949	1 |- ( ph -> H ~~> ( A + B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   + caddc 8095   < clt 8273   - cmin 8409   ZZcz 9540   ZZ>=cuz 9816   RR+crp 9949   abscabs 11637   ~~> cli 11918

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-rp 9950  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919

This theorem is referenced by:  climaddc1  11969  climcvg1nlem  11989  isumadd  12072