Theorem bfplem11 16008

Description: Lemma for bfp 16009. Apply the deduction theorem.

Hypotheses

Ref	Expression
bfp.1	|- X = dom dom M
bfplem.2a	|- M e. CMet
bfplem.4a	|- Y e. X
bfplem.7a	|- K e. RR+
bfplem.8a	|- K < 1

Assertion

Ref	Expression
bfplem11	|- ((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))) -> E!z e. X (F` z) = z)

Distinct variable groups:   x,F,y,z   x,X,y,z   x,Y,y,z   x,M,y,z   x,K,y,z

Proof of Theorem bfplem11

Step	Hyp	Ref	Expression
1		fveq1 4680	. . . 4 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (F` z) = (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` z))
2	1	eqeq1d 1892	. . 3 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> ((F` z) = z <-> (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` z) = z))
3	2	reubidv 2260	. 2 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (E!z e. X (F` z) = z <-> E!z e. X (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` z) = z))
4		bfp.1	. . 3 |- X = dom dom M
5		bfplem.2a	. . 3 |- M e. CMet
6		feq1 4551	. . . . . 6 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (F:X-->X <-> if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})):X-->X))
7		fveq1 4680	. . . . . . . . . 10 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (F` u) = (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u))
8		fveq1 4680	. . . . . . . . . 10 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (F` v) = (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v))
9	7, 8	opreq12d 4900	. . . . . . . . 9 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> ((F` u)M(F` v)) = ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)))
10	9	breq1d 3348	. . . . . . . 8 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (((F` u)M(F` v)) <_ (K x. (uMv)) <-> ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)) <_ (K x. (uMv))))
11	10	2ralbidv 2140	. . . . . . 7 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (A.u e. X A.v e. X ((F` u)M(F` v)) <_ (K x. (uMv)) <-> A.u e. X A.v e. X ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)) <_ (K x. (uMv))))
12		fveq2 4681	. . . . . . . . . 10 |- (x = u -> (F` x) = (F` u))
13	12	opreq1d 4897	. . . . . . . . 9 |- (x = u -> ((F` x)M(F` y)) = ((F` u)M(F` y)))
14		opreq1 4889	. . . . . . . . . 10 |- (x = u -> (xMy) = (uMy))
15	14	opreq2d 4898	. . . . . . . . 9 |- (x = u -> (K x. (xMy)) = (K x. (uMy)))
16	13, 15	breq12d 3351	. . . . . . . 8 |- (x = u -> (((F` x)M(F` y)) <_ (K x. (xMy)) <-> ((F` u)M(F` y)) <_ (K x. (uMy))))
17		fveq2 4681	. . . . . . . . . 10 |- (y = v -> (F` y) = (F` v))
18	17	opreq2d 4898	. . . . . . . . 9 |- (y = v -> ((F` u)M(F` y)) = ((F` u)M(F` v)))
19		opreq2 4890	. . . . . . . . . 10 |- (y = v -> (uMy) = (uMv))
20	19	opreq2d 4898	. . . . . . . . 9 |- (y = v -> (K x. (uMy)) = (K x. (uMv)))
21	18, 20	breq12d 3351	. . . . . . . 8 |- (y = v -> (((F` u)M(F` y)) <_ (K x. (uMy)) <-> ((F` u)M(F` v)) <_ (K x. (uMv))))
22	16, 21	cbvral2v 2283	. . . . . . 7 |- (A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy)) <-> A.u e. X A.v e. X ((F` u)M(F` v)) <_ (K x. (uMv)))
23	11, 22	syl5bb 591	. . . . . 6 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy)) <-> A.u e. X A.v e. X ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)) <_ (K x. (uMv))))
24	6, 23	anbi12d 690	. . . . 5 |- (F = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> ((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))) <-> (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})):X-->X /\ A.u e. X A.v e. X ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)) <_ (K x. (uMv)))))
25		feq1 4551	. . . . . 6 |- ((X X. {Y}) = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> ((X X. {Y}):X-->X <-> if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})):X-->X))
26		fveq1 4680	. . . . . . . . 9 |- ((X X. {Y}) = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> ((X X. {Y})` u) = (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u))
27		fveq1 4680	. . . . . . . . 9 |- ((X X. {Y}) = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> ((X X. {Y})` v) = (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v))
28	26, 27	opreq12d 4900	. . . . . . . 8 |- ((X X. {Y}) = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (((X X. {Y})` u)M((X X. {Y})` v)) = ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)))
29	28	breq1d 3348	. . . . . . 7 |- ((X X. {Y}) = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> ((((X X. {Y})` u)M((X X. {Y})` v)) <_ (K x. (uMv)) <-> ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)) <_ (K x. (uMv))))
30	29	2ralbidv 2140	. . . . . 6 |- ((X X. {Y}) = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (A.u e. X A.v e. X (((X X. {Y})` u)M((X X. {Y})` v)) <_ (K x. (uMv)) <-> A.u e. X A.v e. X ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)) <_ (K x. (uMv))))
31	25, 30	anbi12d 690	. . . . 5 |- ((X X. {Y}) = if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})) -> (((X X. {Y}):X-->X /\ A.u e. X A.v e. X (((X X. {Y})` u)M((X X. {Y})` v)) <_ (K x. (uMv))) <-> (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})):X-->X /\ A.u e. X A.v e. X ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)) <_ (K x. (uMv)))))
32		bfplem.4a	. . . . . . 7 |- Y e. X
33	32	fconst6 15700	. . . . . 6 |- (X X. {Y}):X-->X
34	32	elisseti 2301	. . . . . . . . . . 11 |- Y e. _V
35	34	fvconst2 4822	. . . . . . . . . 10 |- (u e. X -> ((X X. {Y})` u) = Y)
36	34	fvconst2 4822	. . . . . . . . . 10 |- (v e. X -> ((X X. {Y})` v) = Y)
37	35, 36	opreqan12d 4902	. . . . . . . . 9 |- ((u e. X /\ v e. X) -> (((X X. {Y})` u)M((X X. {Y})` v)) = (YMY))
38	5	cmsmeti 9240	. . . . . . . . . 10 |- M e. Met
39	4	met0 9092	. . . . . . . . . 10 |- ((M e. Met /\ Y e. X) -> (YMY) = 0)
40	38, 32, 39	mp2an 761	. . . . . . . . 9 |- (YMY) = 0
41	37, 40	syl6eq 1944	. . . . . . . 8 |- ((u e. X /\ v e. X) -> (((X X. {Y})` u)M((X X. {Y})` v)) = 0)
42		bfplem.7a	. . . . . . . . . . 11 |- K e. RR+
43		rpre 7236	. . . . . . . . . . 11 |- (K e. RR+ -> K e. RR)
44	42, 43	ax-mp 7	. . . . . . . . . 10 |- K e. RR
45	44	a1i 8	. . . . . . . . 9 |- ((u e. X /\ v e. X) -> K e. RR)
46		rpge0 7241	. . . . . . . . . . 11 |- (K e. RR+ -> 0 <_ K)
47	42, 46	ax-mp 7	. . . . . . . . . 10 |- 0 <_ K
48	47	a1i 8	. . . . . . . . 9 |- ((u e. X /\ v e. X) -> 0 <_ K)
49	4	metcl 9088	. . . . . . . . . 10 |- ((M e. Met /\ u e. X /\ v e. X) -> (uMv) e. RR)
50	38, 49	mp3an1 1178	. . . . . . . . 9 |- ((u e. X /\ v e. X) -> (uMv) e. RR)
51	4	metge0 9096	. . . . . . . . . 10 |- ((M e. Met /\ u e. X /\ v e. X) -> 0 <_ (uMv))
52	38, 51	mp3an1 1178	. . . . . . . . 9 |- ((u e. X /\ v e. X) -> 0 <_ (uMv))
53		mulge0 6868	. . . . . . . . 9 |- (((K e. RR /\ 0 <_ K) /\ ((uMv) e. RR /\ 0 <_ (uMv))) -> 0 <_ (K x. (uMv)))
54	45, 48, 50, 52, 53	syl22anc 1101	. . . . . . . 8 |- ((u e. X /\ v e. X) -> 0 <_ (K x. (uMv)))
55	41, 54	eqbrtrd 3357	. . . . . . 7 |- ((u e. X /\ v e. X) -> (((X X. {Y})` u)M((X X. {Y})` v)) <_ (K x. (uMv)))
56	55	rgen2a 2160	. . . . . 6 |- A.u e. X A.v e. X (((X X. {Y})` u)M((X X. {Y})` v)) <_ (K x. (uMv))
57	33, 56	pm3.2i 307	. . . . 5 |- ((X X. {Y}):X-->X /\ A.u e. X A.v e. X (((X X. {Y})` u)M((X X. {Y})` v)) <_ (K x. (uMv)))
58	24, 31, 57	elimhyp 3021	. . . 4 |- (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})):X-->X /\ A.u e. X A.v e. X ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)) <_ (K x. (uMv)))
59	58	simpli 347	. . 3 |- if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y})):X-->X
60		eqid 1884	. . 3 |- ({<.<.a, b>., c>. | ((a e. X /\ b e. X) /\ c = (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` a))} seq1 (NN X. {Y})) = ({<.<.a, b>., c>. | ((a e. X /\ b e. X) /\ c = (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` a))} seq1 (NN X. {Y}))
61		eqid 1884	. . 3 |- (YM(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` Y)) = (YM(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` Y))
62		bfplem.8a	. . 3 |- K < 1
63	58	simpri 351	. . 3 |- A.u e. X A.v e. X ((if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` u)M(if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` v)) <_ (K x. (uMv))
64	4, 5, 59, 32, 60, 61, 42, 62, 63	bfplem10 16007	. 2 |- E!z e. X (if((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))), F, (X X. {Y}))` z) = z
65	3, 64	dedth 3011	1 |- ((F:X-->X /\ A.x e. X A.y e. X ((F` x)M(F` y)) <_ (K x. (xMy))) -> E!z e. X (F` z) = z)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E!wreu 2107  ifcif 2982  {csn 3044   class class class wbr 3338   X. cxp 3984  dom cdm 3986  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  RRcr 6385  0cc0 6386  1c1 6387   x. cmul 6391   <_ cle 6448  NNcn 6449  RR+crp 6453   < clt 6653   seq1 cseq1 7720  Metcme 9066  CMetcms 9199

This theorem is referenced by:  bfp 16009

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-sup 5664  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-div 6892  df-n 7108  df-2 7154  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240  df-met 9070  df-lm 9200  df-cau 9201  df-cmet 9202

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