Theorem climadd 12070

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 12026	. . 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 12026	. . 3 |- ( G ~~> B -> B e. CC )
8	6, 7	syl 14	. 2 |- ( ph -> B e. CC )
9		addcl 8294	. . 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 12054	. . 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 1278	. 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 12053	1 |- ( ph -> H ~~> ( A + B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   + caddc 8172   < clt 8350   - cmin 8487   ZZcz 9623   ZZ>=cuz 9900   RR+crp 10033   abscabs 11741   ~~> cli 12022

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-rp 10034  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023

This theorem is referenced by:  climaddc1  12073  climcvg1nlem  12093  isumadd  12176