Theorem ltxrlt 7823

Description: The standard less-than <RR and the extended real less-than < are identical when restricted to the non-extended reals RR. (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
ltxrlt	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )

Proof of Theorem ltxrlt

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ltxr 7798	. . . . 5 |- < = ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) )
2	1	breqi 3930	. . . 4 |- ( A < B <-> A ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) ) B )
3		brun 3974	. . . 4 |- ( A ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) ) B <-> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B \/ A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B ) )
4	2, 3	bitri 183	. . 3 |- ( A < B <-> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B \/ A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B ) )
5		eleq1 2200	. . . . . . 7 |- ( x = A -> ( x e. RR <-> A e. RR ) )
6		breq1 3927	. . . . . . 7 |- ( x = A -> ( x <RR y <-> A <RR y ) )
7	5, 6	3anbi13d 1292	. . . . . 6 |- ( x = A -> ( ( x e. RR /\ y e. RR /\ x <RR y ) <-> ( A e. RR /\ y e. RR /\ A <RR y ) ) )
8		eleq1 2200	. . . . . . 7 |- ( y = B -> ( y e. RR <-> B e. RR ) )
9		breq2 3928	. . . . . . 7 |- ( y = B -> ( A <RR y <-> A <RR B ) )
10	8, 9	3anbi23d 1293	. . . . . 6 |- ( y = B -> ( ( A e. RR /\ y e. RR /\ A <RR y ) <-> ( A e. RR /\ B e. RR /\ A <RR B ) ) )
11		eqid 2137	. . . . . 6 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } = { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) }
12	7, 10, 11	brabg 4186	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B <-> ( A e. RR /\ B e. RR /\ A <RR B ) ) )
13		simp3 983	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> A <RR B )
14	12, 13	syl6bi 162	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B -> A <RR B ) )
15		brun 3974	. . . . 5 |- ( A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B <-> ( A ( ( RR u. { -oo } ) X. { +oo } ) B \/ A ( { -oo } X. RR ) B ) )
16		brxp 4565	. . . . . . . . . . 11 |- ( A ( ( RR u. { -oo } ) X. { +oo } ) B <-> ( A e. ( RR u. { -oo } ) /\ B e. { +oo } ) )
17	16	simprbi 273	. . . . . . . . . 10 |- ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> B e. { +oo } )
18		elsni 3540	. . . . . . . . . 10 |- ( B e. { +oo } -> B = +oo )
19	17, 18	syl 14	. . . . . . . . 9 |- ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> B = +oo )
20	19	a1i 9	. . . . . . . 8 |- ( B e. RR -> ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> B = +oo ) )
21		renepnf 7806	. . . . . . . . 9 |- ( B e. RR -> B =/= +oo )
22	21	neneqd 2327	. . . . . . . 8 |- ( B e. RR -> -. B = +oo )
23		pm2.24 610	. . . . . . . 8 |- ( B = +oo -> ( -. B = +oo -> A <RR B ) )
24	20, 22, 23	syl6ci 1421	. . . . . . 7 |- ( B e. RR -> ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> A <RR B ) )
25	24	adantl 275	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> A <RR B ) )
26		brxp 4565	. . . . . . . . . . 11 |- ( A ( { -oo } X. RR ) B <-> ( A e. { -oo } /\ B e. RR ) )
27	26	simplbi 272	. . . . . . . . . 10 |- ( A ( { -oo } X. RR ) B -> A e. { -oo } )
28		elsni 3540	. . . . . . . . . 10 |- ( A e. { -oo } -> A = -oo )
29	27, 28	syl 14	. . . . . . . . 9 |- ( A ( { -oo } X. RR ) B -> A = -oo )
30	29	a1i 9	. . . . . . . 8 |- ( A e. RR -> ( A ( { -oo } X. RR ) B -> A = -oo ) )
31		renemnf 7807	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
32	31	neneqd 2327	. . . . . . . 8 |- ( A e. RR -> -. A = -oo )
33		pm2.24 610	. . . . . . . 8 |- ( A = -oo -> ( -. A = -oo -> A <RR B ) )
34	30, 32, 33	syl6ci 1421	. . . . . . 7 |- ( A e. RR -> ( A ( { -oo } X. RR ) B -> A <RR B ) )
35	34	adantr 274	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A ( { -oo } X. RR ) B -> A <RR B ) )
36	25, 35	jaod 706	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( A ( ( RR u. { -oo } ) X. { +oo } ) B \/ A ( { -oo } X. RR ) B ) -> A <RR B ) )
37	15, 36	syl5bi 151	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B -> A <RR B ) )
38	14, 37	jaod 706	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B \/ A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B ) -> A <RR B ) )
39	4, 38	syl5bi 151	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <RR B ) )
40	12	3adant3 1001	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B <-> ( A e. RR /\ B e. RR /\ A <RR B ) ) )
41	40	ibir 176	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B )
42	41	orcd 722	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B \/ A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B ) )
43	42, 4	sylibr 133	. . 3 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> A < B )
44	43	3expia 1183	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A <RR B -> A < B ) )
45	39, 44	impbid 128	1 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   /\ w3a 962   = wceq 1331   e. wcel 1480   u. cun 3064   {csn 3522   class class class wbr 3924   {copab 3983   X. cxp 4532   RRcr 7612   <RR cltrr 7617   +oocpnf 7790   -oocmnf 7791   < clt 7793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-pnf 7795  df-mnf 7796  df-ltxr 7798

This theorem is referenced by:  axltirr  7824  axltwlin  7825  axlttrn  7826  axltadd  7827  axapti  7828  axmulgt0  7829  axsuploc  7830  0lt1  7882  recexre  8333  recexgt0  8335  remulext1  8354  arch  8967  caucvgrelemcau  10745  caucvgre  10746