Theorem hbsum1 6983

Description: Bound-variable hypothesis builder for sum.

Hypothesis

Ref	Expression
hbsum1.1	|- (x e. A -> A.k x e. A)

Assertion

Ref	Expression
hbsum1	|- (x e. sum_k e. A B -> A.k x e. sum_k e. A B)

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

Proof of Theorem hbsum1

Step	Hyp	Ref	Expression
1		df-sum 6980	. 2 |- sum_k e. A B = ({y | E.mE.n e. (ZZ>` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n))} u. U.{y | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y)})
2		ax-17 971	. . . . . 6 |- (n e. (ZZ>` m) -> A.k n e. (ZZ>` m))
3		hbsum1.1	. . . . . . . 8 |- (x e. A -> A.k x e. A)
4		ax-17 971	. . . . . . . 8 |- (x e. (m...n) -> A.k x e. (m...n))
5	3, 4	hbeq 1565	. . . . . . 7 |- (A = (m...n) -> A.k A = (m...n))
6		ax-17 971	. . . . . . . . 9 |- (y e. <.m, + >. -> A.k y e. <.m, + >.)
7		ax-17 971	. . . . . . . . 9 |- (y e. seq -> A.k y e. seq )
8		hbopab1 2813	. . . . . . . . . 10 |- (y e. {<.k, z>. | z = B} -> A.k y e. {<.k, z>. | z = B})
9		ax-17 971	. . . . . . . . . 10 |- (y e. ZZ -> A.k y e. ZZ)
10	8, 9	hbres 3370	. . . . . . . . 9 |- (y e. ({<.k, z>. | z = B} |` ZZ) -> A.k y e. ({<.k, z>. | z = B} |` ZZ))
11	6, 7, 10	hbopr 3981	. . . . . . . 8 |- (y e. (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) -> A.k y e. (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)))
12		ax-17 971	. . . . . . . 8 |- (y e. n -> A.k y e. n)
13	11, 12	hbfv 3729	. . . . . . 7 |- (y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n) -> A.k y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n))
14	5, 13	hban 1009	. . . . . 6 |- ((A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n)) -> A.k(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n)))
15	2, 14	hbrex 1688	. . . . 5 |- (E.n e. (ZZ>` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n)) -> A.kE.n e. (ZZ>` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n)))
16	15	hbex 1006	. . . 4 |- (E.mE.n e. (ZZ>` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n)) -> A.kE.mE.n e. (ZZ>` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n)))
17	16	hbab 1467	. . 3 |- (x e. {y | E.mE.n e. (ZZ>` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n))} -> A.k x e. {y | E.mE.n e. (ZZ>` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n))})
18		ax-17 971	. . . . . 6 |- (m e. ZZ -> A.k m e. ZZ)
19		ax-17 971	. . . . . . . 8 |- (x e. (ZZ>` m) -> A.k x e. (ZZ>` m))
20	3, 19	hbeq 1565	. . . . . . 7 |- (A = (ZZ>` m) -> A.k A = (ZZ>` m))
21		ax-17 971	. . . . . . . . 9 |- (n e. <.m, + >. -> A.k n e. <.m, + >.)
22		ax-17 971	. . . . . . . . 9 |- (n e. seq -> A.k n e. seq )
23		hbopab1 2813	. . . . . . . . . 10 |- (n e. {<.k, z>. | z = B} -> A.k n e. {<.k, z>. | z = B})
24		ax-17 971	. . . . . . . . . 10 |- (n e. ZZ -> A.k n e. ZZ)
25	23, 24	hbres 3370	. . . . . . . . 9 |- (n e. ({<.k, z>. | z = B} |` ZZ) -> A.k n e. ({<.k, z>. | z = B} |` ZZ))
26	21, 22, 25	hbopr 3981	. . . . . . . 8 |- (n e. (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) -> A.k n e. (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)))
27		ax-17 971	. . . . . . . 8 |- (n e. ~~> -> A.k n e. ~~> )
28		ax-17 971	. . . . . . . 8 |- (n e. y -> A.k n e. y)
29	26, 27, 28	hbbr 2658	. . . . . . 7 |- ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y -> A.k(<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y)
30	20, 29	hban 1009	. . . . . 6 |- ((A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y) -> A.k(A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y))
31	18, 30	hbrex 1688	. . . . 5 |- (E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y) -> A.kE.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y))
32	31	hbab 1467	. . . 4 |- (x e. {y | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y)} -> A.k x e. {y | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y)})
33	32	hbuni 2509	. . 3 |- (x e. U.{y | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y)} -> A.k x e. U.{y | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y)})
34	17, 33	hbun 2186	. 2 |- (x e. ({y | E.mE.n e. (ZZ>` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n))} u. U.{y | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y)}) -> A.k x e. ({y | E.mE.n e. (ZZ>` m)(A = (m...n) /\ y e. ((<.m, + >. seq ({<.k, z>. | z = B} |` ZZ))` n))} u. U.{y | E.m e. ZZ (A = (ZZ>` m) /\ (<.m, + >. seq ({<.k, z>. | z = B} |` ZZ)) ~~> y)}))
35	1, 34	hbxfr 1563	1 |- (x e. sum_k e. A B -> A.k x e. sum_k e. A B)

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 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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