Theorem climmullem6 7125

Description: Lemma for climmul 7128.

Hypotheses

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

Assertion

Ref	Expression
climmullem6	|- ((t e. (ZZ>` M) /\ ph) -> ((F` t) e. CC /\ (G` t) e. CC /\ (H` t) = ((F` t) x. (G` t))))

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

Proof of Theorem climmullem6

Step	Hyp	Ref	Expression
1		fveq2 3724	. . . . . 6 |- (k = t -> (F` k) = (F` t))
2	1	eleq1d 1540	. . . . 5 |- (k = t -> ((F` k) e. CC <-> (F` t) e. CC))
3		fveq2 3724	. . . . . 6 |- (k = t -> (G` k) = (G` t))
4	3	eleq1d 1540	. . . . 5 |- (k = t -> ((G` k) e. CC <-> (G` t) e. CC))
5		fveq2 3724	. . . . . 6 |- (k = t -> (H` k) = (H` t))
6	1, 3	opreq12d 3978	. . . . . 6 |- (k = t -> ((F` k) x. (G` k)) = ((F` t) x. (G` t)))
7	5, 6	eqeq12d 1489	. . . . 5 |- (k = t -> ((H` k) = ((F` k) x. (G` k)) <-> (H` t) = ((F` t) x. (G` t))))
8	2, 4, 7	3anbi123d 893	. . . 4 |- (k = t -> (((F` k) e. CC /\ (G` k) e. CC /\ (H` k) = ((F` k) x. (G` k))) <-> ((F` t) e. CC /\ (G` t) e. CC /\ (H` t) = ((F` t) x. (G` t)))))
9	8	rcla4va 1875	. . 3 |- ((t e. (ZZ>` M) /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) e. CC /\ (H` k) = ((F` k) x. (G` k)))) -> ((F` t) e. CC /\ (G` t) e. CC /\ (H` t) = ((F` t) x. (G` t))))
10	9	adantrl 394	. 2 |- ((t e. (ZZ>` M) /\ ((F ~~> A /\ G ~~> B) /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) e. CC /\ (H` k) = ((F` k) x. (G` k))))) -> ((F` t) e. CC /\ (G` t) e. CC /\ (H` t) = ((F` t) x. (G` t))))
11		climmullem.6	. 2 |- (ph <-> ((F ~~> A /\ G ~~> B) /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) e. CC /\ (H` k) = ((F` k) x. (G` k)))))
12	10, 11	sylan2b 452	1 |- ((t e. (ZZ>` M) /\ ph) -> ((F` t) e. CC /\ (G` t) e. CC /\ (H` t) = ((F` t) x. (G` t))))

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   x. cmul 5239  ZZ>cuz 6417   ~~> cli 6974

This theorem is referenced by:  climmullem8 7127

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-3an 777  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-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-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

Copyright terms: Public domain