Theorem fzrevral 7698

Description: Reversal of scanning order inside of a quantification over a finite set of sequential integers.

Assertion

Ref	Expression
fzrevral	|- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.j e. (M...N)ph <-> A.k e. ((K - N)...(K - M))[(K - k) / j]ph))

Distinct variable groups:   j,k,K   j,M,k   j,N,k   ph,k

Proof of Theorem fzrevral

Step	Hyp	Ref	Expression
1		simpr 350	. . . . . . . 8 |- ((((M e. ZZ /\ N e. ZZ) /\ K e. ZZ) /\ k e. ((K - N)...(K - M))) -> k e. ((K - N)...(K - M)))
2		fzrev 7689	. . . . . . . . . 10 |- (((M e. ZZ /\ N e. ZZ) /\ (K e. ZZ /\ k e. ZZ)) -> (k e. ((K - N)...(K - M)) <-> (K - k) e. (M...N)))
3	2	anassrs 489	. . . . . . . . 9 |- ((((M e. ZZ /\ N e. ZZ) /\ K e. ZZ) /\ k e. ZZ) -> (k e. ((K - N)...(K - M)) <-> (K - k) e. (M...N)))
4		elfzelz 7652	. . . . . . . . 9 |- (k e. ((K - N)...(K - M)) -> k e. ZZ)
5	3, 4	sylan2 500	. . . . . . . 8 |- ((((M e. ZZ /\ N e. ZZ) /\ K e. ZZ) /\ k e. ((K - N)...(K - M))) -> (k e. ((K - N)...(K - M)) <-> (K - k) e. (M...N)))
6	1, 5	mpbid 212	. . . . . . 7 |- ((((M e. ZZ /\ N e. ZZ) /\ K e. ZZ) /\ k e. ((K - N)...(K - M))) -> (K - k) e. (M...N))
7		ra4sbc 2536	. . . . . . 7 |- ((K - k) e. (M...N) -> (A.j e. (M...N)ph -> [(K - k) / j]ph))
8	6, 7	syl 12	. . . . . 6 |- ((((M e. ZZ /\ N e. ZZ) /\ K e. ZZ) /\ k e. ((K - N)...(K - M))) -> (A.j e. (M...N)ph -> [(K - k) / j]ph))
9	8	ex 402	. . . . 5 |- (((M e. ZZ /\ N e. ZZ) /\ K e. ZZ) -> (k e. ((K - N)...(K - M)) -> (A.j e. (M...N)ph -> [(K - k) / j]ph)))
10	9	3impa 1062	. . . 4 |- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (k e. ((K - N)...(K - M)) -> (A.j e. (M...N)ph -> [(K - k) / j]ph)))
11	10	com23 36	. . 3 |- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.j e. (M...N)ph -> (k e. ((K - N)...(K - M)) -> [(K - k) / j]ph)))
12	11	r19.21adv 2181	. 2 |- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.j e. (M...N)ph -> A.k e. ((K - N)...(K - M))[(K - k) / j]ph))
13		ax-17 1317	. . . 4 |- ((N e. ZZ /\ K e. ZZ) -> A.j(N e. ZZ /\ K e. ZZ))
14		ax-17 1317	. . . . 5 |- (k e. ((K - N)...(K - M)) -> A.j k e. ((K - N)...(K - M)))
15		oprex 4907	. . . . . 6 |- (K - k) e. _V
16	15	hbsbc1v 2464	. . . . 5 |- ([(K - k) / j]ph -> A.j[(K - k) / j]ph)
17	14, 16	hbral 2146	. . . 4 |- (A.k e. ((K - N)...(K - M))[(K - k) / j]ph -> A.jA.k e. ((K - N)...(K - M))[(K - k) / j]ph)
18		fzrev2i 7691	. . . . . . . . 9 |- ((N e. ZZ /\ K e. ZZ /\ j e. (M...N)) -> (K - j) e. ((K - N)...(K - M)))
19	18	3expa 1067	. . . . . . . 8 |- (((N e. ZZ /\ K e. ZZ) /\ j e. (M...N)) -> (K - j) e. ((K - N)...(K - M)))
20		opreq2 4890	. . . . . . . . . 10 |- (k = (K - j) -> (K - k) = (K - (K - j)))
21		dfsbcq 2455	. . . . . . . . . 10 |- ((K - k) = (K - (K - j)) -> ([(K - k) / j]ph <-> [(K - (K - j)) / j]ph))
22	20, 21	syl 12	. . . . . . . . 9 |- (k = (K - j) -> ([(K - k) / j]ph <-> [(K - (K - j)) / j]ph))
23	22	rcla4v 2376	. . . . . . . 8 |- ((K - j) e. ((K - N)...(K - M)) -> (A.k e. ((K - N)...(K - M))[(K - k) / j]ph -> [(K - (K - j)) / j]ph))
24	19, 23	syl 12	. . . . . . 7 |- (((N e. ZZ /\ K e. ZZ) /\ j e. (M...N)) -> (A.k e. ((K - N)...(K - M))[(K - k) / j]ph -> [(K - (K - j)) / j]ph))
25		nncan 6635	. . . . . . . . . . 11 |- ((K e. CC /\ j e. CC) -> (K - (K - j)) = j)
26		zcn 7349	. . . . . . . . . . 11 |- (K e. ZZ -> K e. CC)
27		elfzelz 7652	. . . . . . . . . . . 12 |- (j e. (M...N) -> j e. ZZ)
28		zcn 7349	. . . . . . . . . . . 12 |- (j e. ZZ -> j e. CC)
29	27, 28	syl 12	. . . . . . . . . . 11 |- (j e. (M...N) -> j e. CC)
30	25, 26, 29	syl2an 503	. . . . . . . . . 10 |- ((K e. ZZ /\ j e. (M...N)) -> (K - (K - j)) = j)
31	30	eqcomd 1889	. . . . . . . . 9 |- ((K e. ZZ /\ j e. (M...N)) -> j = (K - (K - j)))
32		sbceq1a 2456	. . . . . . . . 9 |- (j = (K - (K - j)) -> (ph <-> [(K - (K - j)) / j]ph))
33	31, 32	syl 12	. . . . . . . 8 |- ((K e. ZZ /\ j e. (M...N)) -> (ph <-> [(K - (K - j)) / j]ph))
34	33	adantll 428	. . . . . . 7 |- (((N e. ZZ /\ K e. ZZ) /\ j e. (M...N)) -> (ph <-> [(K - (K - j)) / j]ph))
35	24, 34	sylibrd 221	. . . . . 6 |- (((N e. ZZ /\ K e. ZZ) /\ j e. (M...N)) -> (A.k e. ((K - N)...(K - M))[(K - k) / j]ph -> ph))
36	35	ex 402	. . . . 5 |- ((N e. ZZ /\ K e. ZZ) -> (j e. (M...N) -> (A.k e. ((K - N)...(K - M))[(K - k) / j]ph -> ph)))
37	36	com23 36	. . . 4 |- ((N e. ZZ /\ K e. ZZ) -> (A.k e. ((K - N)...(K - M))[(K - k) / j]ph -> (j e. (M...N) -> ph)))
38	13, 17, 37	r19.21ad 2180	. . 3 |- ((N e. ZZ /\ K e. ZZ) -> (A.k e. ((K - N)...(K - M))[(K - k) / j]ph -> A.j e. (M...N)ph))
39	38	3adant1 894	. 2 |- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.k e. ((K - N)...(K - M))[(K - k) / j]ph -> A.j e. (M...N)ph))
40	12, 39	impbid 574	1 |- ((M e. ZZ /\ N e. ZZ /\ K e. ZZ) -> (A.j e. (M...N)ph <-> A.k e. ((K - N)...(K - M))[(K - k) / j]ph))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  (class class class)co 4884  CCcc 6384   - cmin 6445  ZZcz 6451  ...cfz 7637

This theorem is referenced by:  fzrevral2 7699  fzrevral3 7700  fzshftral 7701  fsumrev 8289  fsumshft 8291

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-13 1311  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  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-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-fz 7638

Copyright terms: Public domain