Theorem eqord2 8246

Description: A strictly decreasing real function on a subset of RR is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.)

Hypotheses

Ref	Expression
ltord.1	|- ( x = y -> A = B )
ltord.2	|- ( x = C -> A = M )
ltord.3	|- ( x = D -> A = N )
ltord.4	|- S C_ RR
ltord.5	|- ( ( ph /\ x e. S ) -> A e. RR )
ltord2.6	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> B < A ) )

Assertion

Ref	Expression
eqord2	|- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) )

Distinct variable groups:   x, B   x, y, C   x, D, y   x, M, y   x, N, y   ph, x, y   x, S, y

Allowed substitution hints:   A( x, y)   B( y)

Proof of Theorem eqord2

Step	Hyp	Ref	Expression
1		ltord.1	. . . 4 |- ( x = y -> A = B )
2	1	negeqd 7957	. . 3 |- ( x = y -> -u A = -u B )
3		ltord.2	. . . 4 |- ( x = C -> A = M )
4	3	negeqd 7957	. . 3 |- ( x = C -> -u A = -u M )
5		ltord.3	. . . 4 |- ( x = D -> A = N )
6	5	negeqd 7957	. . 3 |- ( x = D -> -u A = -u N )
7		ltord.4	. . 3 |- S C_ RR
8		ltord.5	. . . 4 |- ( ( ph /\ x e. S ) -> A e. RR )
9	8	renegcld 8142	. . 3 |- ( ( ph /\ x e. S ) -> -u A e. RR )
10		ltord2.6	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> B < A ) )
11	8	ralrimiva 2505	. . . . . . 7 |- ( ph -> A. x e. S A e. RR )
12	1	eleq1d 2208	. . . . . . . 8 |- ( x = y -> ( A e. RR <-> B e. RR ) )
13	12	rspccva 2788	. . . . . . 7 |- ( ( A. x e. S A e. RR /\ y e. S ) -> B e. RR )
14	11, 13	sylan 281	. . . . . 6 |- ( ( ph /\ y e. S ) -> B e. RR )
15	14	adantrl 469	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> B e. RR )
16	8	adantrr 470	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> A e. RR )
17		ltneg 8224	. . . . 5 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> -u A < -u B ) )
18	15, 16, 17	syl2anc 408	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( B < A <-> -u A < -u B ) )
19	10, 18	sylibd 148	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> -u A < -u B ) )
20	2, 4, 6, 7, 9, 19	eqord1 8245	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> -u M = -u N ) )
21	3	eleq1d 2208	. . . . . . 7 |- ( x = C -> ( A e. RR <-> M e. RR ) )
22	21	rspccva 2788	. . . . . 6 |- ( ( A. x e. S A e. RR /\ C e. S ) -> M e. RR )
23	11, 22	sylan 281	. . . . 5 |- ( ( ph /\ C e. S ) -> M e. RR )
24	23	adantrr 470	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. RR )
25	24	recnd 7794	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. CC )
26	5	eleq1d 2208	. . . . . . 7 |- ( x = D -> ( A e. RR <-> N e. RR ) )
27	26	rspccva 2788	. . . . . 6 |- ( ( A. x e. S A e. RR /\ D e. S ) -> N e. RR )
28	11, 27	sylan 281	. . . . 5 |- ( ( ph /\ D e. S ) -> N e. RR )
29	28	adantrl 469	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. RR )
30	29	recnd 7794	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. CC )
31	25, 30	neg11ad 8069	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( -u M = -u N <-> M = N ) )
32	20, 31	bitrd 187	1 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2416   C_ wss 3071   class class class wbr 3929   RRcr 7619   < clt 7800   -ucneg 7934

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-apti 7735  ax-pre-ltadd 7736

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-ltxr 7805  df-sub 7935  df-neg 7936

This theorem is referenced by: (None)