Theorem iooint 6373

Description: Intersection of two open intervals of extended reals.

Assertion

Ref	Expression
iooint	|- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ D e. RR*)) -> ((A(,)B) i^i (C(,)D)) = (if(A <_ C, C, A)(,)if(B <_ D, B, D)))

Proof of Theorem iooint

Step	Hyp	Ref	Expression
1		xrmaxltt 5915	. . . . . . . . 9 |- ((A e. RR* /\ C e. RR* /\ x e. RR*) -> (if(A <_ C, C, A) < x <-> (A < x /\ C < x)))
2	1	3expa 835	. . . . . . . 8 |- (((A e. RR* /\ C e. RR*) /\ x e. RR*) -> (if(A <_ C, C, A) < x <-> (A < x /\ C < x)))
3	2	adantlr 395	. . . . . . 7 |- ((((A e. RR* /\ C e. RR*) /\ (B e. RR* /\ D e. RR*)) /\ x e. RR*) -> (if(A <_ C, C, A) < x <-> (A < x /\ C < x)))
4		xrltmint 5916	. . . . . . . . . 10 |- ((x e. RR* /\ B e. RR* /\ D e. RR*) -> (x < if(B <_ D, B, D) <-> (x < B /\ x < D)))
5	4	3coml 842	. . . . . . . . 9 |- ((B e. RR* /\ D e. RR* /\ x e. RR*) -> (x < if(B <_ D, B, D) <-> (x < B /\ x < D)))
6	5	3expa 835	. . . . . . . 8 |- (((B e. RR* /\ D e. RR*) /\ x e. RR*) -> (x < if(B <_ D, B, D) <-> (x < B /\ x < D)))
7	6	adantll 394	. . . . . . 7 |- ((((A e. RR* /\ C e. RR*) /\ (B e. RR* /\ D e. RR*)) /\ x e. RR*) -> (x < if(B <_ D, B, D) <-> (x < B /\ x < D)))
8	3, 7	anbi12d 630	. . . . . 6 |- ((((A e. RR* /\ C e. RR*) /\ (B e. RR* /\ D e. RR*)) /\ x e. RR*) -> ((if(A <_ C, C, A) < x /\ x < if(B <_ D, B, D)) <-> ((A < x /\ C < x) /\ (x < B /\ x < D))))
9		an4 508	. . . . . 6 |- (((A < x /\ C < x) /\ (x < B /\ x < D)) <-> ((A < x /\ x < B) /\ (C < x /\ x < D)))
10	8, 9	syl6bb 538	. . . . 5 |- ((((A e. RR* /\ C e. RR*) /\ (B e. RR* /\ D e. RR*)) /\ x e. RR*) -> ((if(A <_ C, C, A) < x /\ x < if(B <_ D, B, D)) <-> ((A < x /\ x < B) /\ (C < x /\ x < D))))
11	10	rabbidv 1809	. . . 4 |- (((A e. RR* /\ C e. RR*) /\ (B e. RR* /\ D e. RR*)) -> {x e. RR* | (if(A <_ C, C, A) < x /\ x < if(B <_ D, B, D))} = {x e. RR* | ((A < x /\ x < B) /\ (C < x /\ x < D))})
12	11	an4s 510	. . 3 |- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ D e. RR*)) -> {x e. RR* | (if(A <_ C, C, A) < x /\ x < if(B <_ D, B, D))} = {x e. RR* | ((A < x /\ x < B) /\ (C < x /\ x < D))})
13		inrab 2274	. . 3 |- ({x e. RR* | (A < x /\ x < B)} i^i {x e. RR* | (C < x /\ x < D)}) = {x e. RR* | ((A < x /\ x < B) /\ (C < x /\ x < D))}
14	12, 13	syl6reqr 1529	. 2 |- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ D e. RR*)) -> ({x e. RR* | (A < x /\ x < B)} i^i {x e. RR* | (C < x /\ x < D)}) = {x e. RR* | (if(A <_ C, C, A) < x /\ x < if(B <_ D, B, D))})
15		ioovalt 6367	. . 3 |- ((A e. RR* /\ B e. RR*) -> (A(,)B) = {x e. RR* | (A < x /\ x < B)})
16		ioovalt 6367	. . 3 |- ((C e. RR* /\ D e. RR*) -> (C(,)D) = {x e. RR* | (C < x /\ x < D)})
17	15, 16	ineqan12d 2222	. 2 |- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ D e. RR*)) -> ((A(,)B) i^i (C(,)D)) = ({x e. RR* | (A < x /\ x < B)} i^i {x e. RR* | (C < x /\ x < D)}))
18		ioovalt 6367	. . 3 |- ((if(A <_ C, C, A) e. RR* /\ if(B <_ D, B, D) e. RR*) -> (if(A <_ C, C, A)(,)if(B <_ D, B, D)) = {x e. RR* | (if(A <_ C, C, A) < x /\ x < if(B <_ D, B, D))})
19		ifcl 2384	. . . . 5 |- ((C e. RR* /\ A e. RR*) -> if(A <_ C, C, A) e. RR*)
20	19	ancoms 438	. . . 4 |- ((A e. RR* /\ C e. RR*) -> if(A <_ C, C, A) e. RR*)
21	20	ad2ant2r 411	. . 3 |- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ D e. RR*)) -> if(A <_ C, C, A) e. RR*)
22		ifcl 2384	. . . 4 |- ((B e. RR* /\ D e. RR*) -> if(B <_ D, B, D) e. RR*)
23	22	ad2ant2l 410	. . 3 |- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ D e. RR*)) -> if(B <_ D, B, D) e. RR*)
24	18, 21, 23	sylanc 473	. 2 |- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ D e. RR*)) -> (if(A <_ C, C, A)(,)if(B <_ D, B, D)) = {x e. RR* | (if(A <_ C, C, A) < x /\ x < if(B <_ D, B, D))})
25	14, 17, 24	3eqtr4d 1520	1 |- (((A e. RR* /\ B e. RR*) /\ (C e. RR* /\ D e. RR*)) -> ((A(,)B) i^i (C(,)D)) = (if(A <_ C, C, A)(,)if(B <_ D, B, D)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  {crab 1651   i^i cin 2049  ifcif 2365   class class class wbr 2624  (class class class)co 3969   <_ cle 5307  RR*cxr 5497   < clt 5498  (,)cioo 6358

This theorem is referenced by:  retopbas 7652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-ltp 5102  df-enr 5178  df-nr 5179  df-ltr 5182  df-0r 5183  df-c 5252  df-r 5256  df-lt 5259  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-ioo 6362

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