Theorem fzrevralt 6459

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

Assertion

Ref	Expression
fzrevralt	|- ((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 fzrevralt

Step	Hyp	Ref	Expression
1		pm3.27 323	. . . . . . . 8 |- ((((M e. ZZ /\ N e. ZZ) /\ K e. ZZ) /\ k e. ((K - N)...(K - M))) -> k e. ((K - N)...(K - M)))
2		fzrevt 6451	. . . . . . . . . 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 441	. . . . . . . . 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 6422	. . . . . . . . 9 |- (k e. ((K - N)...(K - M)) -> k e. ZZ)
5	3, 4	sylan2 451	. . . . . . . 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 195	. . . . . . 7 |- ((((M e. ZZ /\ N e. ZZ) /\ K e. ZZ) /\ k e. ((K - N)...(K - M))) -> (K - k) e. (M...N))
7		ra4sbc 1993	. . . . . . 7 |- ((K - k) e. (M...N) -> (A.j e. (M...N)ph -> [(K - k) / j]ph))
8	6, 7	syl 10	. . . . . 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 373	. . . . 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 827	. . . 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 32	. . 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 1715	. 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-17 969	. . . 4 |- ((N e. ZZ /\ K e. ZZ) -> A.j(N e. ZZ /\ K e. ZZ))
14		ax-17 969	. . . . 5 |- (k e. ((K - N)...(K - M)) -> A.j k e. ((K - N)...(K - M)))
15		oprex 3974	. . . . . 6 |- (K - k) e. V
16	15	hbsbc1v 1946	. . . . 5 |- ([(K - k) / j]ph -> A.j[(K - k) / j]ph)
17	14, 16	hbral 1683	. . . 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		fzrev2it 6453	. . . . . . . . 9 |- ((N e. ZZ /\ K e. ZZ /\ j e. (M...N)) -> (K - j) e. ((K - N)...(K - M)))
19	18	3expa 832	. . . . . . . 8 |- (((N e. ZZ /\ K e. ZZ) /\ j e. (M...N)) -> (K - j) e. ((K - N)...(K - M)))
20		opreq2 3960	. . . . . . . . . 10 |- (k = (K - j) -> (K - k) = (K - (K - j)))
21		dfsbcq 1939	. . . . . . . . . 10 |- ((K - k) = (K - (K - j)) -> ([(K - k) / j]ph <-> [(K - (K - j)) / j]ph))
22	20, 21	syl 10	. . . . . . . . 9 |- (k = (K - j) -> ([(K - k) / j]ph <-> [(K - (K - j)) / j]ph))
23	22	rcla4v 1869	. . . . . . . 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 10	. . . . . . 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		nncant 5449	. . . . . . . . . . 11 |- ((K e. CC /\ j e. CC) -> (K - (K - j)) = j)
26		zcnt 6095	. . . . . . . . . . 11 |- (K e. ZZ -> K e. CC)
27		elfzelz 6422	. . . . . . . . . . . 12 |- (j e. (M...N) -> j e. ZZ)
28		zcnt 6095	. . . . . . . . . . . 12 |- (j e. ZZ -> j e. CC)
29	27, 28	syl 10	. . . . . . . . . . 11 |- (j e. (M...N) -> j e. CC)
30	25, 26, 29	syl2an 454	. . . . . . . . . 10 |- ((K e. ZZ /\ j e. (M...N)) -> (K - (K - j)) = j)
31	30	eqcomd 1477	. . . . . . . . 9 |- ((K e. ZZ /\ j e. (M...N)) -> j = (K - (K - j)))
32		sbceq1a 1940	. . . . . . . . 9 |- (j = (K - (K - j)) -> (ph <-> [(K - (K - j)) / j]ph))
33	31, 32	syl 10	. . . . . . . 8 |- ((K e. ZZ /\ j e. (M...N)) -> (ph <-> [(K - (K - j)) / j]ph))
34	33	adantll 392	. . . . . . 7 |- (((N e. ZZ /\ K e. ZZ) /\ j e. (M...N)) -> (ph <-> [(K - (K - j)) / j]ph))
35	24, 34	sylibrd 204	. . . . . 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 373	. . . . 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 32	. . . 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 1714	. . 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 796	. 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 515	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 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  [wsbc 1168  A.wral 1642  (class class class)co 3954  CCcc 5212   - cmin 5272  ZZcz 5278  ...cfz 6407

This theorem is referenced by:  fzrevral2t 6460  fzrevral3t 6461  fzshftralt 6462  fsumrev 6975  fsumshft 6977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-n 5881  df-n0 6055  df-z 6091  df-fz 6408

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