Theorem ltxrlt 8212

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 8186	. . . . 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 4089	. . . 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 4135	. . . 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 184	. . 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 2292	. . . . . . 7 |- ( x = A -> ( x e. RR <-> A e. RR ) )
6		breq1 4086	. . . . . . 7 |- ( x = A -> ( x <RR y <-> A <RR y ) )
7	5, 6	3anbi13d 1348	. . . . . 6 |- ( x = A -> ( ( x e. RR /\ y e. RR /\ x <RR y ) <-> ( A e. RR /\ y e. RR /\ A <RR y ) ) )
8		eleq1 2292	. . . . . . 7 |- ( y = B -> ( y e. RR <-> B e. RR ) )
9		breq2 4087	. . . . . . 7 |- ( y = B -> ( A <RR y <-> A <RR B ) )
10	8, 9	3anbi23d 1349	. . . . . 6 |- ( y = B -> ( ( A e. RR /\ y e. RR /\ A <RR y ) <-> ( A e. RR /\ B e. RR /\ A <RR B ) ) )
11		eqid 2229	. . . . . 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 4357	. . . . 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 1023	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> A <RR B )
14	12, 13	biimtrdi 163	. . . 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 4135	. . . . 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 4750	. . . . . . . . . . 11 |- ( A ( ( RR u. { -oo } ) X. { +oo } ) B <-> ( A e. ( RR u. { -oo } ) /\ B e. { +oo } ) )
17	16	simprbi 275	. . . . . . . . . 10 |- ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> B e. { +oo } )
18		elsni 3684	. . . . . . . . . 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 8194	. . . . . . . . 9 |- ( B e. RR -> B =/= +oo )
22	21	neneqd 2421	. . . . . . . 8 |- ( B e. RR -> -. B = +oo )
23		pm2.24 624	. . . . . . . 8 |- ( B = +oo -> ( -. B = +oo -> A <RR B ) )
24	20, 22, 23	syl6ci 1488	. . . . . . 7 |- ( B e. RR -> ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> A <RR B ) )
25	24	adantl 277	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> A <RR B ) )
26		brxp 4750	. . . . . . . . . . 11 |- ( A ( { -oo } X. RR ) B <-> ( A e. { -oo } /\ B e. RR ) )
27	26	simplbi 274	. . . . . . . . . 10 |- ( A ( { -oo } X. RR ) B -> A e. { -oo } )
28		elsni 3684	. . . . . . . . . 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 8195	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
32	31	neneqd 2421	. . . . . . . 8 |- ( A e. RR -> -. A = -oo )
33		pm2.24 624	. . . . . . . 8 |- ( A = -oo -> ( -. A = -oo -> A <RR B ) )
34	30, 32, 33	syl6ci 1488	. . . . . . 7 |- ( A e. RR -> ( A ( { -oo } X. RR ) B -> A <RR B ) )
35	34	adantr 276	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A ( { -oo } X. RR ) B -> A <RR B ) )
36	25, 35	jaod 722	. . . . 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	biimtrid 152	. . . 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 722	. . 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	biimtrid 152	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <RR B ) )
40	12	3adant3 1041	. . . . . 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 177	. . . . 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 738	. . . 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 134	. . 3 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> A < B )
44	43	3expia 1229	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A <RR B -> A < B ) )
45	39, 44	impbid 129	1 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   u. cun 3195   {csn 3666   class class class wbr 4083   {copab 4144   X. cxp 4717   RRcr 7998   <RR cltrr 8003   +oocpnf 8178   -oocmnf 8179   < clt 8181

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8183  df-mnf 8184  df-ltxr 8186

This theorem is referenced by:  axltirr  8213  axltwlin  8214  axlttrn  8215  axltadd  8216  axapti  8217  axmulgt0  8218  axsuploc  8219  0lt1  8273  recexre  8725  recexgt0  8727  remulext1  8746  arch  9366  caucvgrelemcau  11491  caucvgre  11492