Theorem fdc1 15813

Description: Variant of fdc 15812 with no specified base value.

Hypotheses

Ref	Expression
fdc1.1	|- A e. _V
fdc1.2	|- M e. ZZ
fdc1.3	|- Z = (ZZ>=` M)
fdc1.4	|- N = (M + 1)
fdc1.5	|- (a = (f` M) -> (ze <-> si))
fdc1.6	|- (a = (f` (k - 1)) -> (ph <-> ps))
fdc1.7	|- (b = (f` k) -> (ps <-> ch))
fdc1.8	|- (a = (f` n) -> (th <-> ta))
fdc1.9	|- (et -> E.a e. A ze)
fdc1.10	|- (et -> R Fr A)
fdc1.11	|- ((et /\ a e. A) -> (th \/ E.b e. A ph))
fdc1.12	|- (((et /\ ph) /\ (a e. A /\ b e. A)) -> bRa)

Assertion

Ref	Expression
fdc1	|- (et -> E.n e. Z E.f(f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch))

Distinct variable groups:   A,a,b,f,n   R,a,b   M,a,b,f,k,n   Z,a,b,n   N,a,b,f,k,n   ph,f,k   ps,a   ch,a,b,n   th,f,n   ta,a,b   et,a,b,f,n   ze,b,f,n   si,a

Proof of Theorem fdc1

Step	Hyp	Ref	Expression
1		fdc1.9	. 2 |- (et -> E.a e. A ze)
2		eleq1 1957	. . . . . . . 8 |- (c = a -> (c e. A <-> a e. A))
3	2	anbi2d 678	. . . . . . 7 |- (c = a -> ((et /\ c e. A) <-> (et /\ a e. A)))
4		sbequ12r 1546	. . . . . . 7 |- (c = a -> ([c / a]ze <-> ze))
5	3, 4	anbi12d 690	. . . . . 6 |- (c = a -> (((et /\ c e. A) /\ [c / a]ze) <-> ((et /\ a e. A) /\ ze)))
6	5	imbi1d 675	. . . . 5 |- (c = a -> ((((et /\ c e. A) /\ [c / a]ze) -> E.n e. Z E.f(f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch)) <-> (((et /\ a e. A) /\ ze) -> E.n e. Z E.f(f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch))))
7		fdc1.1	. . . . . . . 8 |- A e. _V
8		fdc1.2	. . . . . . . 8 |- M e. ZZ
9		fdc1.3	. . . . . . . 8 |- Z = (ZZ>=` M)
10		fdc1.4	. . . . . . . 8 |- N = (M + 1)
11		ax-17 1317	. . . . . . . . 9 |- (ps -> A.aps)
12		fdc1.6	. . . . . . . . 9 |- (a = (f` (k - 1)) -> (ph <-> ps))
13	11, 12	sbhypf 2452	. . . . . . . 8 |- (d = (f` (k - 1)) -> ([d / a]ph <-> ps))
14		fdc1.7	. . . . . . . 8 |- (b = (f` k) -> (ps <-> ch))
15		ax-17 1317	. . . . . . . . 9 |- (ta -> A.ata)
16		fdc1.8	. . . . . . . . 9 |- (a = (f` n) -> (th <-> ta))
17	15, 16	sbhypf 2452	. . . . . . . 8 |- (d = (f` n) -> ([d / a]th <-> ta))
18		simprl 450	. . . . . . . 8 |- ((et /\ (c e. A /\ [c / a]ze)) -> c e. A)
19		fdc1.10	. . . . . . . . 9 |- (et -> R Fr A)
20	19	adantr 425	. . . . . . . 8 |- ((et /\ (c e. A /\ [c / a]ze)) -> R Fr A)
21		ax-17 1317	. . . . . . . . . . 11 |- ((et /\ d e. A) -> A.a(et /\ d e. A))
22		ax-17 1317	. . . . . . . . . . . . 13 |- (th -> A.dth)
23	22	hbsb3 1575	. . . . . . . . . . . 12 |- ([d / a]th -> A.a[d / a]th)
24		ax-17 1317	. . . . . . . . . . . . 13 |- (b e. A -> A.a b e. A)
25		ax-17 1317	. . . . . . . . . . . . . 14 |- (ph -> A.dph)
26	25	hbsb3 1575	. . . . . . . . . . . . 13 |- ([d / a]ph -> A.a[d / a]ph)
27	24, 26	hbrex 2149	. . . . . . . . . . . 12 |- (E.b e. A [d / a]ph -> A.aE.b e. A [d / a]ph)
28	23, 27	hbor 1355	. . . . . . . . . . 11 |- (([d / a]th \/ E.b e. A [d / a]ph) -> A.a([d / a]th \/ E.b e. A [d / a]ph))
29	21, 28	hbim 1354	. . . . . . . . . 10 |- (((et /\ d e. A) -> ([d / a]th \/ E.b e. A [d / a]ph)) -> A.a((et /\ d e. A) -> ([d / a]th \/ E.b e. A [d / a]ph)))
30		eleq1 1957	. . . . . . . . . . . 12 |- (a = d -> (a e. A <-> d e. A))
31	30	anbi2d 678	. . . . . . . . . . 11 |- (a = d -> ((et /\ a e. A) <-> (et /\ d e. A)))
32		sbequ12 1545	. . . . . . . . . . . 12 |- (a = d -> (th <-> [d / a]th))
33		sbequ12 1545	. . . . . . . . . . . . 13 |- (a = d -> (ph <-> [d / a]ph))
34	33	rexbidv 2124	. . . . . . . . . . . 12 |- (a = d -> (E.b e. A ph <-> E.b e. A [d / a]ph))
35	32, 34	orbi12d 689	. . . . . . . . . . 11 |- (a = d -> ((th \/ E.b e. A ph) <-> ([d / a]th \/ E.b e. A [d / a]ph)))
36	31, 35	imbi12d 688	. . . . . . . . . 10 |- (a = d -> (((et /\ a e. A) -> (th \/ E.b e. A ph)) <-> ((et /\ d e. A) -> ([d / a]th \/ E.b e. A [d / a]ph))))
37		fdc1.11	. . . . . . . . . 10 |- ((et /\ a e. A) -> (th \/ E.b e. A ph))
38	29, 36, 37	chvar 1530	. . . . . . . . 9 |- ((et /\ d e. A) -> ([d / a]th \/ E.b e. A [d / a]ph))
39	38	adantlr 429	. . . . . . . 8 |- (((et /\ (c e. A /\ [c / a]ze)) /\ d e. A) -> ([d / a]th \/ E.b e. A [d / a]ph))
40		ax-17 1317	. . . . . . . . . . . . 13 |- (et -> A.aet)
41	40, 26	hban 1356	. . . . . . . . . . . 12 |- ((et /\ [d / a]ph) -> A.a(et /\ [d / a]ph))
42		ax-17 1317	. . . . . . . . . . . 12 |- ((d e. A /\ b e. A) -> A.a(d e. A /\ b e. A))
43	41, 42	hban 1356	. . . . . . . . . . 11 |- (((et /\ [d / a]ph) /\ (d e. A /\ b e. A)) -> A.a((et /\ [d / a]ph) /\ (d e. A /\ b e. A)))
44		ax-17 1317	. . . . . . . . . . 11 |- (bRd -> A.a bRd)
45	43, 44	hbim 1354	. . . . . . . . . 10 |- ((((et /\ [d / a]ph) /\ (d e. A /\ b e. A)) -> bRd) -> A.a(((et /\ [d / a]ph) /\ (d e. A /\ b e. A)) -> bRd))
46	33	anbi2d 678	. . . . . . . . . . . 12 |- (a = d -> ((et /\ ph) <-> (et /\ [d / a]ph)))
47	30	anbi1d 679	. . . . . . . . . . . 12 |- (a = d -> ((a e. A /\ b e. A) <-> (d e. A /\ b e. A)))
48	46, 47	anbi12d 690	. . . . . . . . . . 11 |- (a = d -> (((et /\ ph) /\ (a e. A /\ b e. A)) <-> ((et /\ [d / a]ph) /\ (d e. A /\ b e. A))))
49		breq2 3342	. . . . . . . . . . 11 |- (a = d -> (bRa <-> bRd))
50	48, 49	imbi12d 688	. . . . . . . . . 10 |- (a = d -> ((((et /\ ph) /\ (a e. A /\ b e. A)) -> bRa) <-> (((et /\ [d / a]ph) /\ (d e. A /\ b e. A)) -> bRd)))
51		fdc1.12	. . . . . . . . . 10 |- (((et /\ ph) /\ (a e. A /\ b e. A)) -> bRa)
52	45, 50, 51	chvar 1530	. . . . . . . . 9 |- (((et /\ [d / a]ph) /\ (d e. A /\ b e. A)) -> bRd)
53	52	adantllr 433	. . . . . . . 8 |- ((((et /\ (c e. A /\ [c / a]ze)) /\ [d / a]ph) /\ (d e. A /\ b e. A)) -> bRd)
54	7, 8, 9, 10, 13, 14, 17, 18, 20, 39, 53	fdc 15812	. . . . . . 7 |- ((et /\ (c e. A /\ [c / a]ze)) -> E.n e. Z E.f(f:(M...n)-->A /\ ((f` M) = c /\ ta) /\ A.k e. (N...n)ch))
55	54	anassrs 489	. . . . . 6 |- (((et /\ c e. A) /\ [c / a]ze) -> E.n e. Z E.f(f:(M...n)-->A /\ ((f` M) = c /\ ta) /\ A.k e. (N...n)ch))
56		idd 75	. . . . . . . . 9 |- (((et /\ c e. A) /\ [c / a]ze) -> (f:(M...n)-->A -> f:(M...n)-->A))
57		dfsbcq 2455	. . . . . . . . . . . . 13 |- ((f` M) = c -> ([(f` M) / a]ze <-> [c / a]ze))
58		fvex 4689	. . . . . . . . . . . . . 14 |- (f` M) e. _V
59		fdc1.5	. . . . . . . . . . . . . 14 |- (a = (f` M) -> (ze <-> si))
60	58, 59	sbcie 2485	. . . . . . . . . . . . 13 |- ([(f` M) / a]ze <-> si)
61	57, 60	syl5rbbr 594	. . . . . . . . . . . 12 |- ((f` M) = c -> ([c / a]ze <-> si))
62	61	biimpcd 172	. . . . . . . . . . 11 |- ([c / a]ze -> ((f` M) = c -> si))
63	62	adantl 424	. . . . . . . . . 10 |- (((et /\ c e. A) /\ [c / a]ze) -> ((f` M) = c -> si))
64	63	anim1d 619	. . . . . . . . 9 |- (((et /\ c e. A) /\ [c / a]ze) -> (((f` M) = c /\ ta) -> (si /\ ta)))
65		idd 75	. . . . . . . . 9 |- (((et /\ c e. A) /\ [c / a]ze) -> (A.k e. (N...n)ch -> A.k e. (N...n)ch))
66	56, 64, 65	3anim123d 1175	. . . . . . . 8 |- (((et /\ c e. A) /\ [c / a]ze) -> ((f:(M...n)-->A /\ ((f` M) = c /\ ta) /\ A.k e. (N...n)ch) -> (f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch)))
67	66	eximdv 1669	. . . . . . 7 |- (((et /\ c e. A) /\ [c / a]ze) -> (E.f(f:(M...n)-->A /\ ((f` M) = c /\ ta) /\ A.k e. (N...n)ch) -> E.f(f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch)))
68	67	reximdv 2202	. . . . . 6 |- (((et /\ c e. A) /\ [c / a]ze) -> (E.n e. Z E.f(f:(M...n)-->A /\ ((f` M) = c /\ ta) /\ A.k e. (N...n)ch) -> E.n e. Z E.f(f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch)))
69	55, 68	mpd 29	. . . . 5 |- (((et /\ c e. A) /\ [c / a]ze) -> E.n e. Z E.f(f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch))
70	6, 69	chvarv 1712	. . . 4 |- (((et /\ a e. A) /\ ze) -> E.n e. Z E.f(f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch))
71	70	ex 402	. . 3 |- ((et /\ a e. A) -> (ze -> E.n e. Z E.f(f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch)))
72	71	r19.23adva 2216	. 2 |- (et -> (E.a e. A ze -> E.n e. Z E.f(f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch)))
73	1, 72	mpd 29	1 |- (et -> E.n e. Z E.f(f:(M...n)-->A /\ (si /\ ta) /\ A.k e. (N...n)ch))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  A.wral 2105  E.wrex 2106  _Vcvv 2292   class class class wbr 3338   Fr wfr 3623  -->wf 3994  ` cfv 3998  (class class class)co 4884  1c1 6387   + caddc 6389   - cmin 6445  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637

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-uz 7587  df-fz 7638

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