Theorem cbvprodi 14662

Description: Change bound variable in a finite composite.

Hypotheses

Ref	Expression
cbvprod.1	|- (x e. B -> A.k x e. B)
cbvprod.2	|- (x e. C -> A.j x e. C)
cbvprod.3	|- (j = k -> B = C)

Assertion

Ref	Expression
cbvprodi	|- prod_j e. AGB = prod_k e. AGC

Distinct variable groups:   x,A   x,B   x,C   x,G   x,j   x,k

Proof of Theorem cbvprodi

Step	Hyp	Ref	Expression
1		biid 187	. . 3 |- (A = (/) <-> A = (/))
2		cbvprod.1	. . . . . . . . . . . . . 14 |- (x e. B -> A.k x e. B)
3	2	hbeleq 1997	. . . . . . . . . . . . 13 |- (x = B -> A.k x = B)
4		cbvprod.2	. . . . . . . . . . . . . 14 |- (x e. C -> A.j x e. C)
5	4	hbeleq 1997	. . . . . . . . . . . . 13 |- (x = C -> A.j x = C)
6		cbvprod.3	. . . . . . . . . . . . . 14 |- (j = k -> B = C)
7	6	eqeq2d 1895	. . . . . . . . . . . . 13 |- (j = k -> (x = B <-> x = C))
8	3, 5, 7	cbvopab1 3405	. . . . . . . . . . . 12 |- {<.j, x>. | x = B} = {<.k, x>. | x = C}
9		reseq1 4218	. . . . . . . . . . . 12 |- ({<.j, x>. | x = B} = {<.k, x>. | x = C} -> ({<.j, x>. | x = B} |` ZZ) = ({<.k, x>. | x = C} |` ZZ))
10	8, 9	ax-mp 7	. . . . . . . . . . 11 |- ({<.j, x>. | x = B} |` ZZ) = ({<.k, x>. | x = C} |` ZZ)
11	10	opreq2i 4893	. . . . . . . . . 10 |- (<.m, G>. seq ({<.j, x>. | x = B} |` ZZ)) = (<.m, G>. seq ({<.k, x>. | x = C} |` ZZ))
12	11	fveq1i 4682	. . . . . . . . 9 |- ((<.m, G>. seq ({<.j, x>. | x = B} |` ZZ))` n) = ((<.m, G>. seq ({<.k, x>. | x = C} |` ZZ))` n)
13	12	eleq2i 1961	. . . . . . . 8 |- (y e. ((<.m, G>. seq ({<.j, x>. | x = B} |` ZZ))` n) <-> y e. ((<.m, G>. seq ({<.k, x>. | x = C} |` ZZ))` n))
14	13	anbi2i 538	. . . . . . 7 |- ((A = (m...n) /\ y e. ((<.m, G>. seq ({<.j, x>. | x = B} |` ZZ))` n)) <-> (A = (m...n) /\ y e. ((<.m, G>. seq ({<.k, x>. | x = C} |` ZZ))` n)))
15	14	rexbii 2128	. . . . . 6 |- (E.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, G>. seq ({<.j, x>. | x = B} |` ZZ))` n)) <-> E.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, G>. seq ({<.k, x>. | x = C} |` ZZ))` n)))
16	15	exbii 1398	. . . . 5 |- (E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, G>. seq ({<.j, x>. | x = B} |` ZZ))` n)) <-> E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, G>. seq ({<.k, x>. | x = C} |` ZZ))` n)))
17	16	abbii 2006	. . . 4 |- {y | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, G>. seq ({<.j, x>. | x = B} |` ZZ))` n))} = {y | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, G>. seq ({<.k, x>. | x = C} |` ZZ))` n))}
18		df-prod 14653	. . . 4 |- prod3 j e. AGB = {y | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, G>. seq ({<.j, x>. | x = B} |` ZZ))` n))}
19		df-prod 14653	. . . 4 |- prod3 k e. AGC = {y | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, G>. seq ({<.k, x>. | x = C} |` ZZ))` n))}
20	17, 18, 19	3eqtr4i 1921	. . 3 |- prod3 j e. AGB = prod3 k e. AGC
21	1, 20	ifbieq2i 2997	. 2 |- if(A = (/), (Id` G), prod3 j e. AGB) = if(A = (/), (Id` G), prod3 k e. AGC)
22		df-prod2 14655	. 2 |- prod_j e. AGB = if(A = (/), (Id` G), prod3 j e. AGB)
23		df-prod2 14655	. 2 |- prod_k e. AGC = if(A = (/), (Id` G), prod3 k e. AGC)
24	21, 22, 23	3eqtr4i 1921	1 |- prod_j e. AGB = prod_k e. AGC

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  E.wrex 2106  (/)c0 2875  ifcif 2982  <.cop 3046  {copab 3395   |` cres 3988  ` cfv 3998  (class class class)co 4884  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637   seq cseqz 7774  Idcgi 9312   prod3 cprd 14652  prod_cprd2 14654

This theorem is referenced by:  fprodserzfi 14672  fprod1fi 14675  fprodp1fi 14680

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-prod 14653  df-prod2 14655

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