Theorem csbfsum 6965

Description: Distribute substitution for classes over a finite sum.

Assertion

Ref	Expression
csbfsum	|- ((A e. C /\ N e. (ZZ>` M) /\ A.k e. (M...N)[_A / x]_B e. CC) -> [_A / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_A / x]_B)

Distinct variable groups:   A,k   x,k,M   x,N,k

Proof of Theorem csbfsum

Step	Hyp	Ref	Expression
1		csbeq1 1993	. . . . . . 7 |- (y = A -> [_y / x]_B = [_A / x]_B)
2	1	eleq1d 1532	. . . . . 6 |- (y = A -> ([_y / x]_B e. CC <-> [_A / x]_B e. CC))
3	2	ralbidv 1655	. . . . 5 |- (y = A -> (A.k e. (M...N)[_y / x]_B e. CC <-> A.k e. (M...N)[_A / x]_B e. CC))
4	3	anbi2d 614	. . . 4 |- (y = A -> ((N e. (ZZ>` M) /\ A.k e. (M...N)[_y / x]_B e. CC) <-> (N e. (ZZ>` M) /\ A.k e. (M...N)[_A / x]_B e. CC)))
5		csbeq1 1993	. . . . 5 |- (y = A -> [_y / x]_sum_k e. (M...N)B = [_A / x]_sum_k e. (M...N)B)
6	1	sumeq2sdv 6931	. . . . 5 |- (y = A -> sum_k e. (M...N)[_y / x]_B = sum_k e. (M...N)[_A / x]_B)
7	5, 6	eqeq12d 1481	. . . 4 |- (y = A -> ([_y / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_y / x]_B <-> [_A / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_A / x]_B))
8	4, 7	imbi12d 624	. . 3 |- (y = A -> (((N e. (ZZ>` M) /\ A.k e. (M...N)[_y / x]_B e. CC) -> [_y / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_y / x]_B) <-> ((N e. (ZZ>` M) /\ A.k e. (M...N)[_A / x]_B e. CC) -> [_A / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_A / x]_B)))
9		visset 1804	. . . . . 6 |- y e. V
10		oprex 3968	. . . . . 6 |- (B + 0) e. V
11	9, 10	csbfsumlem 6964	. . . . 5 |- (N e. (ZZ>` M) -> [_y / x]_sum_k e. (M...N)(B + 0) = sum_k e. (M...N)[_y / x]_(B + 0))
12	11	adantr 389	. . . 4 |- ((N e. (ZZ>` M) /\ A.k e. (M...N)[_y / x]_B e. CC) -> [_y / x]_sum_k e. (M...N)(B + 0) = sum_k e. (M...N)[_y / x]_(B + 0))
13		ax0id 5253	. . . . . . . . 9 |- (B e. CC -> (B + 0) = B)
14	13	r19.20si 1698	. . . . . . . 8 |- (A.k e. (M...N)B e. CC -> A.k e. (M...N)(B + 0) = B)
15	14	sumeq2d 6929	. . . . . . 7 |- (A.k e. (M...N)B e. CC -> sum_k e. (M...N)(B + 0) = sum_k e. (M...N)B)
16	15	sbimi 1169	. . . . . 6 |- ([y / x]A.k e. (M...N)B e. CC -> [y / x]sum_k e. (M...N)(B + 0) = sum_k e. (M...N)B)
17		sbcralg 1984	. . . . . . . 8 |- (y e. V -> ([y / x]A.k e. (M...N)B e. CC <-> A.k e. (M...N)[y / x]B e. CC))
18	9, 17	ax-mp 7	. . . . . . 7 |- ([y / x]A.k e. (M...N)B e. CC <-> A.k e. (M...N)[y / x]B e. CC)
19		sbcel1g 2003	. . . . . . . . 9 |- (y e. V -> ([y / x]B e. CC <-> [_y / x]_B e. CC))
20	9, 19	ax-mp 7	. . . . . . . 8 |- ([y / x]B e. CC <-> [_y / x]_B e. CC)
21	20	ralbii 1659	. . . . . . 7 |- (A.k e. (M...N)[y / x]B e. CC <-> A.k e. (M...N)[_y / x]_B e. CC)
22	18, 21	bitr 173	. . . . . 6 |- ([y / x]A.k e. (M...N)B e. CC <-> A.k e. (M...N)[_y / x]_B e. CC)
23		sbceqdig 2002	. . . . . . 7 |- (y e. V -> ([y / x]sum_k e. (M...N)(B + 0) = sum_k e. (M...N)B <-> [_y / x]_sum_k e. (M...N)(B + 0) = [_y / x]_sum_k e. (M...N)B))
24	9, 23	ax-mp 7	. . . . . 6 |- ([y / x]sum_k e. (M...N)(B + 0) = sum_k e. (M...N)B <-> [_y / x]_sum_k e. (M...N)(B + 0) = [_y / x]_sum_k e. (M...N)B)
25	16, 22, 24	3imtr3 218	. . . . 5 |- (A.k e. (M...N)[_y / x]_B e. CC -> [_y / x]_sum_k e. (M...N)(B + 0) = [_y / x]_sum_k e. (M...N)B)
26	25	adantl 388	. . . 4 |- ((N e. (ZZ>` M) /\ A.k e. (M...N)[_y / x]_B e. CC) -> [_y / x]_sum_k e. (M...N)(B + 0) = [_y / x]_sum_k e. (M...N)B)
27		ax0id 5253	. . . . . . . 8 |- ([_y / x]_B e. CC -> ([_y / x]_B + 0) = [_y / x]_B)
28		csbopr1g 3973	. . . . . . . . 9 |- (y e. V -> [_y / x]_(B + 0) = ([_y / x]_B + 0))
29	9, 28	ax-mp 7	. . . . . . . 8 |- [_y / x]_(B + 0) = ([_y / x]_B + 0)
30	27, 29	syl5eq 1511	. . . . . . 7 |- ([_y / x]_B e. CC -> [_y / x]_(B + 0) = [_y / x]_B)
31	30	r19.20si 1698	. . . . . 6 |- (A.k e. (M...N)[_y / x]_B e. CC -> A.k e. (M...N)[_y / x]_(B + 0) = [_y / x]_B)
32	31	sumeq2d 6929	. . . . 5 |- (A.k e. (M...N)[_y / x]_B e. CC -> sum_k e. (M...N)[_y / x]_(B + 0) = sum_k e. (M...N)[_y / x]_B)
33	32	adantl 388	. . . 4 |- ((N e. (ZZ>` M) /\ A.k e. (M...N)[_y / x]_B e. CC) -> sum_k e. (M...N)[_y / x]_(B + 0) = sum_k e. (M...N)[_y / x]_B)
34	12, 26, 33	3eqtr3d 1507	. . 3 |- ((N e. (ZZ>` M) /\ A.k e. (M...N)[_y / x]_B e. CC) -> [_y / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_y / x]_B)
35	8, 34	vtoclg 1838	. 2 |- (A e. C -> ((N e. (ZZ>` M) /\ A.k e. (M...N)[_A / x]_B e. CC) -> [_A / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_A / x]_B))
36	35	3impib 829	1 |- ((A e. C /\ N e. (ZZ>` M) /\ A.k e. (M...N)[_A / x]_B e. CC) -> [_A / x]_sum_k e. (M...N)B = sum_k e. (M...N)[_A / x]_B)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  [wsbc 1166  A.wral 1637  Vcvv 1802  [_csb 1991  ` cfv 3172  (class class class)co 3948  CCcc 5204  0cc0 5206   + caddc 5209  ZZ>cuz 6349  ...cfz 6399  sum_csu 6917

This theorem is referenced by:  fsumcom 6966

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-n 5873  df-n0 6047  df-z 6083  df-seq1 6245  df-shft 6278  df-uz 6350  df-fz 6400  df-seqz 6465  df-sum 6918

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