Theorem cbvsumi 8246

Description: Change bound variable in a sum.

Hypotheses

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

Assertion

Ref	Expression
cbvsumi	|- sum_j e. A B = sum_k e. A C

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

Proof of Theorem cbvsumi

Step	Hyp	Ref	Expression
1		cbvsum.1	. . . . . . . . . . . . 13 |- (x e. B -> A.k x e. B)
2	1	hbeleq 1997	. . . . . . . . . . . 12 |- (x = B -> A.k x = B)
3		cbvsum.2	. . . . . . . . . . . . 13 |- (x e. C -> A.j x e. C)
4	3	hbeleq 1997	. . . . . . . . . . . 12 |- (x = C -> A.j x = C)
5		cbvsum.3	. . . . . . . . . . . . 13 |- (j = k -> B = C)
6	5	eqeq2d 1895	. . . . . . . . . . . 12 |- (j = k -> (x = B <-> x = C))
7	2, 4, 6	cbvopab1 3405	. . . . . . . . . . 11 |- {<.j, x>. | x = B} = {<.k, x>. | x = C}
8		reseq1 4218	. . . . . . . . . . 11 |- ({<.j, x>. | x = B} = {<.k, x>. | x = C} -> ({<.j, x>. | x = B} |` ZZ) = ({<.k, x>. | x = C} |` ZZ))
9	7, 8	ax-mp 7	. . . . . . . . . 10 |- ({<.j, x>. | x = B} |` ZZ) = ({<.k, x>. | x = C} |` ZZ)
10	9	opreq2i 4893	. . . . . . . . 9 |- (<.m, + >. seq ({<.j, x>. | x = B} |` ZZ)) = (<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))
11	10	fveq1i 4682	. . . . . . . 8 |- ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ))` n) = ((<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))` n)
12	11	eleq2i 1961	. . . . . . 7 |- (y e. ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ))` n) <-> y e. ((<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))` n))
13	12	anbi2i 538	. . . . . 6 |- ((A = (m...n) /\ y e. ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ))` n)) <-> (A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))` n)))
14	13	rexbii 2128	. . . . 5 |- (E.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ))` n)) <-> E.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))` n)))
15	14	exbii 1398	. . . 4 |- (E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ))` n)) <-> E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))` n)))
16	15	abbii 2006	. . 3 |- {y | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ))` n))} = {y | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))` n))}
17	10	breq1i 3345	. . . . . . 7 |- ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ)) ~~> y <-> (<.m, + >. seq ({<.k, x>. | x = C} |` ZZ)) ~~> y)
18	17	anbi2i 538	. . . . . 6 |- ((A = (ZZ>=` m) /\ (<.m, + >. seq ({<.j, x>. | x = B} |` ZZ)) ~~> y) <-> (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, x>. | x = C} |` ZZ)) ~~> y))
19	18	rexbii 2128	. . . . 5 |- (E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.j, x>. | x = B} |` ZZ)) ~~> y) <-> E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, x>. | x = C} |` ZZ)) ~~> y))
20	19	abbii 2006	. . . 4 |- {y | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.j, x>. | x = B} |` ZZ)) ~~> y)} = {y | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, x>. | x = C} |` ZZ)) ~~> y)}
21	20	unieqi 3187	. . 3 |- U.{y | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.j, x>. | x = B} |` ZZ)) ~~> y)} = U.{y | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, x>. | x = C} |` ZZ)) ~~> y)}
22	16, 21	uneq12i 2753	. 2 |- ({y | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ))` n))} u. U.{y | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.j, x>. | x = B} |` ZZ)) ~~> y)}) = ({y | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))` n))} u. U.{y | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, x>. | x = C} |` ZZ)) ~~> y)})
23		df-sum 8240	. 2 |- sum_j e. A B = ({y | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ))` n))} u. U.{y | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.j, x>. | x = B} |` ZZ)) ~~> y)})
24		df-sum 8240	. 2 |- sum_k e. A C = ({y | E.mE.n e. (ZZ>=` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))` n))} u. U.{y | E.m e. ZZ (A = (ZZ>=` m) /\ (<.m, + >. seq ({<.k, x>. | x = C} |` ZZ)) ~~> y)})
25	22, 23, 24	3eqtr4i 1921	1 |- sum_j e. A B = sum_k e. A C

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  E.wrex 2106   u. cun 2591  <.cop 3046  U.cuni 3177   class class class wbr 3338  {copab 3395   |` cres 3988  ` cfv 3998  (class class class)co 4884   + caddc 6389  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637   seq cseqz 7774   ~~> cli 8234  sum_csu 8239

This theorem is referenced by:  fsumserzfi 8260  fsum1fi 8267  fsump1fi 8271  binomlem2 8327  isumvaltfi 8454  isumnn0nnai 8469  isummulc1 8473  isummulc1ai 8475  arisumi 8487  fsum0diag2 8521  cntrsetlem 14999  isumclf 15828  mettrifi2 15848

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-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-sum 8240

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