Theorem climaddlem2 7115

Description: Lemma for climadd 7117.

Hypotheses

Ref	Expression
climadd.1	|- F e. V
climadd.2	|- G e. V
climadd.3	|- H e. V
climadd.4	|- A e. V
climadd.5	|- B e. V
climaddlem.6	|- (ph <-> ((F ~~> A /\ G ~~> B) /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) e. CC /\ (H` k) = ((F` k) + (G` k)))))

Assertion

Ref	Expression
climaddlem2	|- (ph -> (A e. CC /\ B e. CC))

Distinct variable groups:   k,F   k,G   k,H   k,M

Proof of Theorem climaddlem2

Step	Hyp	Ref	Expression
1		climaddlem.6	. 2 |- (ph <-> ((F ~~> A /\ G ~~> B) /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) e. CC /\ (H` k) = ((F` k) + (G` k)))))
2		climadd.4	. . . . 5 |- A e. V
3		climcl 6978	. . . . 5 |- ((A e. V /\ F ~~> A) -> A e. CC)
4	2, 3	mpan 695	. . . 4 |- (F ~~> A -> A e. CC)
5		climadd.5	. . . . 5 |- B e. V
6		climcl 6978	. . . . 5 |- ((B e. V /\ G ~~> B) -> B e. CC)
7	5, 6	mpan 695	. . . 4 |- (G ~~> B -> B e. CC)
8	4, 7	anim12i 333	. . 3 |- ((F ~~> A /\ G ~~> B) -> (A e. CC /\ B e. CC))
9	8	adantr 389	. 2 |- (((F ~~> A /\ G ~~> B) /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) e. CC /\ (H` k) = ((F` k) + (G` k)))) -> (A e. CC /\ B e. CC))
10	1, 9	sylbi 199	1 |- (ph -> (A e. CC /\ B e. CC))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811   class class class wbr 2619  ` cfv 3182  (class class class)co 3963  CCcc 5232   + caddc 5237  ZZ>cuz 6417   ~~> cli 6974

This theorem is referenced by:  climaddlem3 7116

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965  df-clim 6975

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