Theorem dvidlemap 15414

Description: Lemma for dvid 15418 and dvconst 15417. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)

Hypotheses

Ref	Expression
dvidlem.1	|- ( ph -> F : CC --> CC )
dvidlemap.2	|- ( ( ph /\ ( x e. CC /\ z e. CC /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
dvidlem.3	|- B e. CC

Assertion

Ref	Expression
dvidlemap	|- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

Distinct variable groups:   x, z, B   x, F, z   ph, x, z

Proof of Theorem dvidlemap

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvidlem.1	. . . . . 6 |- ( ph -> F : CC --> CC )
2		cnex 8155	. . . . . . 7 |- CC e. _V
3	2, 2	fpm 6849	. . . . . 6 |- ( F : CC --> CC -> F e. ( CC ^pm CC ) )
4	1, 3	syl 14	. . . . 5 |- ( ph -> F e. ( CC ^pm CC ) )
5		dvfcnpm 15413	. . . . 5 |- ( F e. ( CC ^pm CC ) -> ( CC _D F ) : dom ( CC _D F ) --> CC )
6	4, 5	syl 14	. . . 4 |- ( ph -> ( CC _D F ) : dom ( CC _D F ) --> CC )
7		ssidd 3248	. . . . . . 7 |- ( ph -> CC C_ CC )
8	7, 1, 7	dvbss 15408	. . . . . 6 |- ( ph -> dom ( CC _D F ) C_ CC )
9		reldvg 15402	. . . . . . . . 9 |- ( ( CC C_ CC /\ F e. ( CC ^pm CC ) ) -> Rel ( CC _D F ) )
10	7, 4, 9	syl2anc 411	. . . . . . . 8 |- ( ph -> Rel ( CC _D F ) )
11	10	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> Rel ( CC _D F ) )
12		simpr 110	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> x e. CC )
13		eqid 2231	. . . . . . . . . . 11 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
14	13	cntoptop 15256	. . . . . . . . . 10 |- ( MetOpen ` ( abs o. - ) ) e. Top
15	13	cntoptopon 15255	. . . . . . . . . . . 12 |- ( MetOpen ` ( abs o. - ) ) e. (TopOn ` CC )
16	15	toponunii 14740	. . . . . . . . . . 11 |- CC = U. ( MetOpen ` ( abs o. - ) )
17	16	ntrtop 14851	. . . . . . . . . 10 |- ( ( MetOpen ` ( abs o. - ) ) e. Top -> ( ( int ` ( MetOpen ` ( abs o. - ) ) ) ` CC ) = CC )
18	14, 17	ax-mp 5	. . . . . . . . 9 |- ( ( int ` ( MetOpen ` ( abs o. - ) ) ) ` CC ) = CC
19	12, 18	eleqtrrdi 2325	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> x e. ( ( int ` ( MetOpen ` ( abs o. - ) ) ) ` CC ) )
20		limcresi 15389	. . . . . . . . . 10 |- ( ( z e. CC |-> B ) lim CC x ) C_ ( ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) lim CC x )
21		dvidlem.3	. . . . . . . . . . . 12 |- B e. CC
22		ssidd 3248	. . . . . . . . . . . 12 |- ( ( ph /\ x e. CC ) -> CC C_ CC )
23		cncfmptc 15319	. . . . . . . . . . . 12 |- ( ( B e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
24	21, 22, 22, 23	mp3an2i 1378	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> ( z e. CC |-> B ) e. ( CC -cn-> CC ) )
25		eqidd 2232	. . . . . . . . . . 11 |- ( z = x -> B = B )
26	24, 12, 25	cnmptlimc 15397	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. CC |-> B ) lim CC x ) )
27	20, 26	sselid 3225	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> B e. ( ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) lim CC x ) )
28		breq1 4091	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
29	28	elrab 2962	. . . . . . . . . . . . 13 |- ( z e. { w e. CC | w # x } <-> ( z e. CC /\ z # x ) )
30		dvidlemap.2	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. CC /\ z e. CC /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
31	30	3exp2 1251	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. CC -> ( z e. CC -> ( z # x -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B ) ) ) )
32	31	imp43 355	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. CC ) /\ ( z e. CC /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
33	29, 32	sylan2b 287	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. CC ) /\ z e. { w e. CC | w # x } ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
34	33	mpteq2dva 4179	. . . . . . . . . . 11 |- ( ( ph /\ x e. CC ) -> ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. CC | w # x } |-> B ) )
35		ssrab2 3312	. . . . . . . . . . . 12 |- { w e. CC | w # x } C_ CC
36		resmpt 5061	. . . . . . . . . . . 12 |- ( { w e. CC | w # x } C_ CC -> ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) = ( z e. { w e. CC | w # x } |-> B ) )
37	35, 36	ax-mp 5	. . . . . . . . . . 11 |- ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) = ( z e. { w e. CC | w # x } |-> B )
38	34, 37	eqtr4di 2282	. . . . . . . . . 10 |- ( ( ph /\ x e. CC ) -> ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) )
39	38	oveq1d 6032	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> ( ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) = ( ( ( z e. CC |-> B ) |` { w e. CC | w # x } ) lim CC x ) )
40	27, 39	eleqtrrd 2311	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> B e. ( ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
41	15	toponrestid 14744	. . . . . . . . 9 |- ( MetOpen ` ( abs o. - ) ) = ( ( MetOpen ` ( abs o. - ) ) ↾t CC )
42		eqid 2231	. . . . . . . . 9 |- ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
43	1	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. CC ) -> F : CC --> CC )
44	41, 13, 42, 22, 43, 22	eldvap 15405	. . . . . . . 8 |- ( ( ph /\ x e. CC ) -> ( x ( CC _D F ) B <-> ( x e. ( ( int ` ( MetOpen ` ( abs o. - ) ) ) ` CC ) /\ B e. ( ( z e. { w e. CC | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) ) )
45	19, 40, 44	mpbir2and 952	. . . . . . 7 |- ( ( ph /\ x e. CC ) -> x ( CC _D F ) B )
46		releldm 4967	. . . . . . 7 |- ( ( Rel ( CC _D F ) /\ x ( CC _D F ) B ) -> x e. dom ( CC _D F ) )
47	11, 45, 46	syl2anc 411	. . . . . 6 |- ( ( ph /\ x e. CC ) -> x e. dom ( CC _D F ) )
48	8, 47	eqelssd 3246	. . . . 5 |- ( ph -> dom ( CC _D F ) = CC )
49	48	feq2d 5470	. . . 4 |- ( ph -> ( ( CC _D F ) : dom ( CC _D F ) --> CC <-> ( CC _D F ) : CC --> CC ) )
50	6, 49	mpbid 147	. . 3 |- ( ph -> ( CC _D F ) : CC --> CC )
51	50	ffnd 5483	. 2 |- ( ph -> ( CC _D F ) Fn CC )
52		fnconstg 5534	. . 3 |- ( B e. CC -> ( CC X. { B } ) Fn CC )
53	21, 52	mp1i 10	. 2 |- ( ph -> ( CC X. { B } ) Fn CC )
54	6	adantr 276	. . . . . 6 |- ( ( ph /\ x e. CC ) -> ( CC _D F ) : dom ( CC _D F ) --> CC )
55	54	ffund 5486	. . . . 5 |- ( ( ph /\ x e. CC ) -> Fun ( CC _D F ) )
56		funbrfvb 5686	. . . . 5 |- ( ( Fun ( CC _D F ) /\ x e. dom ( CC _D F ) ) -> ( ( ( CC _D F ) ` x ) = B <-> x ( CC _D F ) B ) )
57	55, 47, 56	syl2anc 411	. . . 4 |- ( ( ph /\ x e. CC ) -> ( ( ( CC _D F ) ` x ) = B <-> x ( CC _D F ) B ) )
58	45, 57	mpbird 167	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = B )
59	21	a1i 9	. . . 4 |- ( ph -> B e. CC )
60		fvconst2g 5867	. . . 4 |- ( ( B e. CC /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
61	59, 60	sylan 283	. . 3 |- ( ( ph /\ x e. CC ) -> ( ( CC X. { B } ) ` x ) = B )
62	58, 61	eqtr4d 2267	. 2 |- ( ( ph /\ x e. CC ) -> ( ( CC _D F ) ` x ) = ( ( CC X. { B } ) ` x ) )
63	51, 53, 62	eqfnfvd 5747	1 |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   {crab 2514   C_ wss 3200   {csn 3669   class class class wbr 4088   |-> cmpt 4150   X. cxp 4723   dom cdm 4725   |` cres 4727   o. ccom 4729   Rel wrel 4730   Fun wfun 5320   Fn wfn 5321   -->wf 5322   ` cfv 5326  (class class class)co 6017   ^pm cpm 6817   CCcc 8029   - cmin 8349   # cap 8760   / cdiv 8851   abscabs 11557   MetOpencmopn 14554   Topctop 14720   intcnt 14816   -cn->ccncf 15293   lim CC climc 15377   _D cdv 15378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-map 6818  df-pm 6819  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-xneg 10006  df-xadd 10007  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-rest 13323  df-topgen 13342  df-psmet 14556  df-xmet 14557  df-met 14558  df-bl 14559  df-mopn 14560  df-top 14721  df-topon 14734  df-bases 14766  df-ntr 14819  df-cn 14911  df-cnp 14912  df-cncf 15294  df-limced 15379  df-dvap 15380

This theorem is referenced by:  dvconst  15417  dvid  15418