Theorem cbvsum 6986

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
cbvsum	|- sum_j e. A B = sum_k e. A C

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

Proof of Theorem cbvsum

Step	Hyp	Ref	Expression
1		cbvsum.1	. . . . . . . . . . . . 13 |- (x e. B -> A.k x e. B)
2	1	hbeleq 1567	. . . . . . . . . . . 12 |- (x = B -> A.k x = B)
3		cbvsum.2	. . . . . . . . . . . . 13 |- (x e. C -> A.j x e. C)
4	3	hbeleq 1567	. . . . . . . . . . . 12 |- (x = C -> A.j x = C)
5		cbvsum.3	. . . . . . . . . . . . 13 |- (j = k -> B = C)
6	5	eqeq2d 1486	. . . . . . . . . . . 12 |- (j = k -> (x = B <-> x = C))
7	2, 4, 6	cbvopab1 2674	. . . . . . . . . . 11 |- {<.j, x>. | x = B} = {<.k, x>. | x = C}
8		reseq1 3368	. . . . . . . . . . 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 3972	. . . . . . . . 9 |- (<.m, + >. seq ({<.j, x>. | x = B} |` ZZ)) = (<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))
11	10	fveq1i 3725	. . . . . . . 8 |- ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ))` n) = ((<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))` n)
12	11	eleq2i 1538	. . . . . . 7 |- (y e. ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ))` n) <-> y e. ((<.m, + >. seq ({<.k, x>. | x = C} |` ZZ))` n))
13	12	anbi2i 480	. . . . . 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 1668	. . . . 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 1051	. . . 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 1575	. . 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 2626	. . . . . . 7 |- ((<.m, + >. seq ({<.j, x>. | x = B} |` ZZ)) ~~> y <-> (<.m, + >. seq ({<.k, x>. | x = C} |` ZZ)) ~~> y)
18	17	anbi2i 480	. . . . . 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 1668	. . . . 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 1575	. . . 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 2511	. . 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 2182	. 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 6980	. 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 6980	. 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	3eqtr4 1505	1 |- sum_j e. A B = sum_k e. A C

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646   u. cun 2045  <.cop 2411  U.cuni 2503   class class class wbr 2619  {copab 2666   |` cres 3172  ` cfv 3182  (class class class)co 3963   + caddc 5237  ZZcz 5298  ZZ>cuz 6417  ...cfz 6467   seq cseqz 6531   ~~> cli 6974  sum_csu 6979

This theorem is referenced by:  fsumserzf 7000  fsum1f 7007  fsump1f 7011  binomlem2 7067  isumvaltf 7193  isumnn0nna 7208  isummulc1 7212  isummulc1a 7214  fnsmnt 7226  fsum0diag2 7259

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-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-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965  df-sum 6980

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