Theorem fzrevralt 6519

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 6511	. . . . . . . . . 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 6482	. . . . . . . . 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 1997	. . . . . . 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 828	. . . 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 1718	. 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-17 971	. . . 4 |- ((N e. ZZ /\ K e. ZZ) -> A.j(N e. ZZ /\ K e. ZZ))
14		ax-17 971	. . . . 5 |- (k e. ((K - N)...(K - M)) -> A.j k e. ((K - N)...(K - M)))
15		oprex 3983	. . . . . 6 |- (K - k) e. V
16	15	hbsbc1v 1950	. . . . 5 |- ([(K - k) / j]ph -> A.j[(K - k) / j]ph)
17	14, 16	hbral 1686	. . . 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 6513	. . . . . . . . 9 |- ((N e. ZZ /\ K e. ZZ /\ j e. (M...N)) -> (K - j) e. ((K - N)...(K - M)))
19	18	3expa 833	. . . . . . . 8 |- (((N e. ZZ /\ K e. ZZ) /\ j e. (M...N)) -> (K - j) e. ((K - N)...(K - M)))
20		opreq2 3969	. . . . . . . . . 10 |- (k = (K - j) -> (K - k) = (K - (K - j)))
21		dfsbcq 1943	. . . . . . . . . 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 1873	. . . . . . . 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 5469	. . . . . . . . . . 11 |- ((K e. CC /\ j e. CC) -> (K - (K - j)) = j)
26		zcnt 6140	. . . . . . . . . . 11 |- (K e. ZZ -> K e. CC)
27		elfzelz 6482	. . . . . . . . . . . 12 |- (j e. (M...N) -> j e. ZZ)
28		zcnt 6140	. . . . . . . . . . . 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 1480	. . . . . . . . 9 |- ((K e. ZZ /\ j e. (M...N)) -> j = (K - (K - j)))
32		sbceq1a 1944	. . . . . . . . 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 1717	. . 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 797	. 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 516	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 775   = wceq 956   e. wcel 958  [wsbc 1170  A.wral 1645  (class class class)co 3963  CCcc 5232   - cmin 5292  ZZcz 5298  ...cfz 6467

This theorem is referenced by:  fzrevral2t 6520  fzrevral3t 6521  fzshftralt 6522  fsumrev 7029  fsumshft 7031

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-9 965  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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  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-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-n 5925  df-n0 6100  df-z 6136  df-fz 6468

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