Theorem iccdificc 39766

Description: The difference of two closed intervals with the same lower bound. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
iccdificc.a	|- ( ph -> A e. RR* )
iccdificc.b	|- ( ph -> B e. RR* )
iccdificc.c	|- ( ph -> C e. RR* )
iccdificc.4	|- ( ph -> A <_ B )

Assertion

Ref	Expression
iccdificc	|- ( ph -> ( ( A [,] C ) \ ( A [,] B ) ) = ( B (,] C ) )

Proof of Theorem iccdificc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccdificc.b	. . . . . 6 |- ( ph -> B e. RR* )
2	1	adantr 481	. . . . 5 |- ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) -> B e. RR* )
3		iccdificc.c	. . . . . 6 |- ( ph -> C e. RR* )
4	3	adantr 481	. . . . 5 |- ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) -> C e. RR* )
5		iccssxr 12256	. . . . . . 7 |- ( A [,] C ) C_ RR*
6		eldifi 3732	. . . . . . 7 |- ( x e. ( ( A [,] C ) \ ( A [,] B ) ) -> x e. ( A [,] C ) )
7	5, 6	sseldi 3601	. . . . . 6 |- ( x e. ( ( A [,] C ) \ ( A [,] B ) ) -> x e. RR* )
8	7	adantl 482	. . . . 5 |- ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) -> x e. RR* )
9		iccdificc.a	. . . . . . . 8 |- ( ph -> A e. RR* )
10	9	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) /\ -. B < x ) -> A e. RR* )
11	2	adantr 481	. . . . . . 7 |- ( ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) /\ -. B < x ) -> B e. RR* )
12	8	adantr 481	. . . . . . 7 |- ( ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) /\ -. B < x ) -> x e. RR* )
13	9	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) -> A e. RR* )
14	6	adantl 482	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) -> x e. ( A [,] C ) )
15		iccgelb 12230	. . . . . . . . 9 |- ( ( A e. RR* /\ C e. RR* /\ x e. ( A [,] C ) ) -> A <_ x )
16	13, 4, 14, 15	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) -> A <_ x )
17	16	adantr 481	. . . . . . 7 |- ( ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) /\ -. B < x ) -> A <_ x )
18		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) /\ -. B < x ) -> -. B < x )
19	8, 2	xrlenltd 10104	. . . . . . . . 9 |- ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) -> ( x <_ B <-> -. B < x ) )
20	19	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) /\ -. B < x ) -> ( x <_ B <-> -. B < x ) )
21	18, 20	mpbird 247	. . . . . . 7 |- ( ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) /\ -. B < x ) -> x <_ B )
22	10, 11, 12, 17, 21	eliccxrd 39753	. . . . . 6 |- ( ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) /\ -. B < x ) -> x e. ( A [,] B ) )
23		eldifn 3733	. . . . . . 7 |- ( x e. ( ( A [,] C ) \ ( A [,] B ) ) -> -. x e. ( A [,] B ) )
24	23	ad2antlr 763	. . . . . 6 |- ( ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) /\ -. B < x ) -> -. x e. ( A [,] B ) )
25	22, 24	condan 835	. . . . 5 |- ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) -> B < x )
26		iccleub 12229	. . . . . 6 |- ( ( A e. RR* /\ C e. RR* /\ x e. ( A [,] C ) ) -> x <_ C )
27	13, 4, 14, 26	syl3anc 1326	. . . . 5 |- ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) -> x <_ C )
28	2, 4, 8, 25, 27	eliocd 39730	. . . 4 |- ( ( ph /\ x e. ( ( A [,] C ) \ ( A [,] B ) ) ) -> x e. ( B (,] C ) )
29	28	ralrimiva 2966	. . 3 |- ( ph -> A. x e. ( ( A [,] C ) \ ( A [,] B ) ) x e. ( B (,] C ) )
30		dfss3 3592	. . 3 |- ( ( ( A [,] C ) \ ( A [,] B ) ) C_ ( B (,] C ) <-> A. x e. ( ( A [,] C ) \ ( A [,] B ) ) x e. ( B (,] C ) )
31	29, 30	sylibr 224	. 2 |- ( ph -> ( ( A [,] C ) \ ( A [,] B ) ) C_ ( B (,] C ) )
32	9	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( B (,] C ) ) -> A e. RR* )
33	3	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( B (,] C ) ) -> C e. RR* )
34		iocssxr 12257	. . . . . . . 8 |- ( B (,] C ) C_ RR*
35		id 22	. . . . . . . 8 |- ( x e. ( B (,] C ) -> x e. ( B (,] C ) )
36	34, 35	sseldi 3601	. . . . . . 7 |- ( x e. ( B (,] C ) -> x e. RR* )
37	36	adantl 482	. . . . . 6 |- ( ( ph /\ x e. ( B (,] C ) ) -> x e. RR* )
38	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( B (,] C ) ) -> B e. RR* )
39		iccdificc.4	. . . . . . . . 9 |- ( ph -> A <_ B )
40	39	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( B (,] C ) ) -> A <_ B )
41		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. ( B (,] C ) ) -> x e. ( B (,] C ) )
42		iocgtlb 39724	. . . . . . . . 9 |- ( ( B e. RR* /\ C e. RR* /\ x e. ( B (,] C ) ) -> B < x )
43	38, 33, 41, 42	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ x e. ( B (,] C ) ) -> B < x )
44	32, 38, 37, 40, 43	xrlelttrd 11991	. . . . . . 7 |- ( ( ph /\ x e. ( B (,] C ) ) -> A < x )
45	32, 37, 44	xrltled 39486	. . . . . 6 |- ( ( ph /\ x e. ( B (,] C ) ) -> A <_ x )
46		iocleub 39725	. . . . . . 7 |- ( ( B e. RR* /\ C e. RR* /\ x e. ( B (,] C ) ) -> x <_ C )
47	38, 33, 41, 46	syl3anc 1326	. . . . . 6 |- ( ( ph /\ x e. ( B (,] C ) ) -> x <_ C )
48	32, 33, 37, 45, 47	eliccxrd 39753	. . . . 5 |- ( ( ph /\ x e. ( B (,] C ) ) -> x e. ( A [,] C ) )
49	32, 38, 37, 43	xrgtnelicc 39765	. . . . 5 |- ( ( ph /\ x e. ( B (,] C ) ) -> -. x e. ( A [,] B ) )
50	48, 49	eldifd 3585	. . . 4 |- ( ( ph /\ x e. ( B (,] C ) ) -> x e. ( ( A [,] C ) \ ( A [,] B ) ) )
51	50	ralrimiva 2966	. . 3 |- ( ph -> A. x e. ( B (,] C ) x e. ( ( A [,] C ) \ ( A [,] B ) ) )
52		dfss3 3592	. . 3 |- ( ( B (,] C ) C_ ( ( A [,] C ) \ ( A [,] B ) ) <-> A. x e. ( B (,] C ) x e. ( ( A [,] C ) \ ( A [,] B ) ) )
53	51, 52	sylibr 224	. 2 |- ( ph -> ( B (,] C ) C_ ( ( A [,] C ) \ ( A [,] B ) ) )
54	31, 53	eqssd 3620	1 |- ( ph -> ( ( A [,] C ) \ ( A [,] B ) ) = ( B (,] C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   RR*cxr 10073   < clt 10074   <_ cle 10075   (,]cioc 12176   [,]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-ioc 12180  df-icc 12182

This theorem is referenced by:  salexct2  40557