Theorem hbsum 6984

Description: Bound-variable hypothesis builder for sum: if x is (effectively) not free in A and B, it is not free in sum_k e. AB.

Hypotheses

Ref	Expression
hbsum.1	|- (y e. A -> A.x y e. A)
hbsum.2	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
hbsum	|- (y e. sum_k e. A B -> A.x y e. sum_k e. A B)

Distinct variable groups:   y,A   y,B   x,k,y

Proof of Theorem hbsum

Step	Hyp	Ref	Expression
1		df-sum 6980	. 2 |- sum_k e. A B = ({z | E.mE.n e. (ZZ>` m)(A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n))} u. U.{z | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z)})
2		ax-17 973	. . . . . 6 |- (n e. (ZZ>` m) -> A.x n e. (ZZ>` m))
3		hbsum.1	. . . . . . . 8 |- (y e. A -> A.x y e. A)
4		ax-17 973	. . . . . . . 8 |- (y e. (m...n) -> A.x y e. (m...n))
5	3, 4	hbeq 1568	. . . . . . 7 |- (A = (m...n) -> A.x A = (m...n))
6		ax-17 973	. . . . . . . . 9 |- (z e. <.m, + >. -> A.x z e. <.m, + >.)
7		ax-17 973	. . . . . . . . 9 |- (z e. seq -> A.x z e. seq )
8		ax-17 973	. . . . . . . . . . . 12 |- (y e. w -> A.x y e. w)
9		hbsum.2	. . . . . . . . . . . 12 |- (y e. B -> A.x y e. B)
10	8, 9	hbeq 1568	. . . . . . . . . . 11 |- (w = B -> A.x w = B)
11	10	hbopab 2818	. . . . . . . . . 10 |- (z e. {<.k, w>. | w = B} -> A.x z e. {<.k, w>. | w = B})
12		ax-17 973	. . . . . . . . . 10 |- (z e. ZZ -> A.x z e. ZZ)
13	11, 12	hbres 3376	. . . . . . . . 9 |- (z e. ({<.k, w>. | w = B} |` ZZ) -> A.x z e. ({<.k, w>. | w = B} |` ZZ))
14	6, 7, 13	hbopr 3987	. . . . . . . 8 |- (z e. (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) -> A.x z e. (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)))
15		ax-17 973	. . . . . . . 8 |- (z e. n -> A.x z e. n)
16	14, 15	hbfv 3735	. . . . . . 7 |- (z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n) -> A.x z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n))
17	5, 16	hban 1011	. . . . . 6 |- ((A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n)) -> A.x(A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n)))
18	2, 17	hbrex 1691	. . . . 5 |- (E.n e. (ZZ>` m)(A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n)) -> A.xE.n e. (ZZ>` m)(A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n)))
19	18	hbex 1008	. . . 4 |- (E.mE.n e. (ZZ>` m)(A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n)) -> A.xE.mE.n e. (ZZ>` m)(A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n)))
20	19	hbab 1470	. . 3 |- (y e. {z | E.mE.n e. (ZZ>` m)(A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n))} -> A.x y e. {z | E.mE.n e. (ZZ>` m)(A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n))})
21		ax-17 973	. . . . . 6 |- (m e. ZZ -> A.x m e. ZZ)
22		ax-17 973	. . . . . . . 8 |- (y e. (ZZ>` m) -> A.x y e. (ZZ>` m))
23	3, 22	hbeq 1568	. . . . . . 7 |- (A = (ZZ>` m) -> A.x A = (ZZ>` m))
24		ax-17 973	. . . . . . . . 9 |- (y e. <.m, + >. -> A.x y e. <.m, + >.)
25		ax-17 973	. . . . . . . . 9 |- (y e. seq -> A.x y e. seq )
26	10	hbopab 2818	. . . . . . . . . 10 |- (y e. {<.k, w>. | w = B} -> A.x y e. {<.k, w>. | w = B})
27		ax-17 973	. . . . . . . . . 10 |- (y e. ZZ -> A.x y e. ZZ)
28	26, 27	hbres 3376	. . . . . . . . 9 |- (y e. ({<.k, w>. | w = B} |` ZZ) -> A.x y e. ({<.k, w>. | w = B} |` ZZ))
29	24, 25, 28	hbopr 3987	. . . . . . . 8 |- (y e. (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) -> A.x y e. (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)))
30		ax-17 973	. . . . . . . 8 |- (y e. ~~> -> A.x y e. ~~> )
31		ax-17 973	. . . . . . . 8 |- (y e. z -> A.x y e. z)
32	29, 30, 31	hbbr 2663	. . . . . . 7 |- ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z -> A.x(<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z)
33	23, 32	hban 1011	. . . . . 6 |- ((A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z) -> A.x(A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z))
34	21, 33	hbrex 1691	. . . . 5 |- (E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z) -> A.xE.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z))
35	34	hbab 1470	. . . 4 |- (y e. {z | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z)} -> A.x y e. {z | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z)})
36	35	hbuni 2513	. . 3 |- (y e. U.{z | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z)} -> A.x y e. U.{z | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z)})
37	20, 36	hbun 2189	. 2 |- (y e. ({z | E.mE.n e. (ZZ>` m)(A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n))} u. U.{z | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z)}) -> A.x y e. ({z | E.mE.n e. (ZZ>` m)(A = (m...n) /\ z e. ((<.m, + >. seq ({<.k, w>. | w = B} |` ZZ))` n))} u. U.{z | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, w>. | w = B} |` ZZ)) ~~> z)}))
38	1, 37	hbxfr 1566	1 |- (y e. sum_k e. A B -> A.x y e. sum_k e. A B)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  E.wrex 1649   u. cun 2048  <.cop 2415  U.cuni 2507   class class class wbr 2624  {copab 2671   |` cres 3178  ` cfv 3188  (class class class)co 3969   + caddc 5249  ZZcz 5310  ZZ>cuz 6418  ...cfz 6468   seq cseqz 6532   ~~> cli 6974  sum_csu 6979

This theorem is referenced by:  fsum0diaglem2 7257  fsum0diag 7258  fsum0diag2 7259  fsum0diag4 7261

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971  df-sum 6980

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