Theorem cncfioobdlem 40109

Description: G actually extends F. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
cncfioobdlem.a	|- ( ph -> A e. RR )
cncfioobdlem.b	|- ( ph -> B e. RR )
cncfioobdlem.f	|- ( ph -> F : ( A (,) B ) --> V )
cncfioobdlem.g	|- G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
cncfioobdlem.c	|- ( ph -> C e. ( A (,) B ) )

Assertion

Ref	Expression
cncfioobdlem	|- ( ph -> ( G ` C ) = ( F ` C ) )

Distinct variable groups:   x, A   x, B   x, C   x, F   ph, x

Allowed substitution hints:   R( x)   G( x)   L( x)   V( x)

Proof of Theorem cncfioobdlem

Step	Hyp	Ref	Expression
1		cncfioobdlem.g	. . 3 |- G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
2	1	a1i 11	. 2 |- ( ph -> G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) )
3		cncfioobdlem.a	. . . . . . 7 |- ( ph -> A e. RR )
4	3	adantr 481	. . . . . 6 |- ( ( ph /\ x = C ) -> A e. RR )
5		cncfioobdlem.c	. . . . . . . . . 10 |- ( ph -> C e. ( A (,) B ) )
6	3	rexrd 10089	. . . . . . . . . . 11 |- ( ph -> A e. RR* )
7		cncfioobdlem.b	. . . . . . . . . . . 12 |- ( ph -> B e. RR )
8	7	rexrd 10089	. . . . . . . . . . 11 |- ( ph -> B e. RR* )
9		elioo2 12216	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,) B ) <-> ( C e. RR /\ A < C /\ C < B ) ) )
10	6, 8, 9	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( C e. ( A (,) B ) <-> ( C e. RR /\ A < C /\ C < B ) ) )
11	5, 10	mpbid 222	. . . . . . . . 9 |- ( ph -> ( C e. RR /\ A < C /\ C < B ) )
12	11	simp2d 1074	. . . . . . . 8 |- ( ph -> A < C )
13	12	adantr 481	. . . . . . 7 |- ( ( ph /\ x = C ) -> A < C )
14		eqcom 2629	. . . . . . . . 9 |- ( x = C <-> C = x )
15	14	biimpi 206	. . . . . . . 8 |- ( x = C -> C = x )
16	15	adantl 482	. . . . . . 7 |- ( ( ph /\ x = C ) -> C = x )
17	13, 16	breqtrd 4679	. . . . . 6 |- ( ( ph /\ x = C ) -> A < x )
18	4, 17	gtned 10172	. . . . 5 |- ( ( ph /\ x = C ) -> x =/= A )
19	18	neneqd 2799	. . . 4 |- ( ( ph /\ x = C ) -> -. x = A )
20	19	iffalsed 4097	. . 3 |- ( ( ph /\ x = C ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
21		simpr 477	. . . . . . 7 |- ( ( ph /\ x = C ) -> x = C )
22	5	elioored 39776	. . . . . . . 8 |- ( ph -> C e. RR )
23	22	adantr 481	. . . . . . 7 |- ( ( ph /\ x = C ) -> C e. RR )
24	21, 23	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ x = C ) -> x e. RR )
25	11	simp3d 1075	. . . . . . . 8 |- ( ph -> C < B )
26	25	adantr 481	. . . . . . 7 |- ( ( ph /\ x = C ) -> C < B )
27	21, 26	eqbrtrd 4675	. . . . . 6 |- ( ( ph /\ x = C ) -> x < B )
28	24, 27	ltned 10173	. . . . 5 |- ( ( ph /\ x = C ) -> x =/= B )
29	28	neneqd 2799	. . . 4 |- ( ( ph /\ x = C ) -> -. x = B )
30	29	iffalsed 4097	. . 3 |- ( ( ph /\ x = C ) -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
31	21	fveq2d 6195	. . 3 |- ( ( ph /\ x = C ) -> ( F ` x ) = ( F ` C ) )
32	20, 30, 31	3eqtrd 2660	. 2 |- ( ( ph /\ x = C ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = ( F ` C ) )
33		ioossicc 12259	. . 3 |- ( A (,) B ) C_ ( A [,] B )
34	33, 5	sseldi 3601	. 2 |- ( ph -> C e. ( A [,] B ) )
35		cncfioobdlem.f	. . 3 |- ( ph -> F : ( A (,) B ) --> V )
36	35, 5	ffvelrnd 6360	. 2 |- ( ph -> ( F ` C ) e. V )
37	2, 32, 34, 36	fvmptd 6288	1 |- ( ph -> ( G ` C ) = ( F ` C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   RR*cxr 10073   < clt 10074   (,)cioo 12175   [,]cicc 12178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioo 12179  df-icc 12182

This theorem is referenced by:  cncfioobd  40110