Theorem eqord2 8382

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 8093	. . 3 |- ( x = y -> -u A = -u B )
3		ltord.2	. . . 4 |- ( x = C -> A = M )
4	3	negeqd 8093	. . 3 |- ( x = C -> -u A = -u M )
5		ltord.3	. . . 4 |- ( x = D -> A = N )
6	5	negeqd 8093	. . 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 8278	. . 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 2539	. . . . . . 7 |- ( ph -> A. x e. S A e. RR )
12	1	eleq1d 2235	. . . . . . . 8 |- ( x = y -> ( A e. RR <-> B e. RR ) )
13	12	rspccva 2829	. . . . . . 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 470	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> B e. RR )
16	8	adantrr 471	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> A e. RR )
17		ltneg 8360	. . . . 5 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> -u A < -u B ) )
18	15, 16, 17	syl2anc 409	. . . 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 8381	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> -u M = -u N ) )
21	3	eleq1d 2235	. . . . . . 7 |- ( x = C -> ( A e. RR <-> M e. RR ) )
22	21	rspccva 2829	. . . . . 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 471	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. RR )
25	24	recnd 7927	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. CC )
26	5	eleq1d 2235	. . . . . . 7 |- ( x = D -> ( A e. RR <-> N e. RR ) )
27	26	rspccva 2829	. . . . . 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 470	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. RR )
30	29	recnd 7927	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. CC )
31	25, 30	neg11ad 8205	. 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 1343   e. wcel 2136   A.wral 2444   C_ wss 3116   class class class wbr 3982   RRcr 7752   < clt 7933   -ucneg 8070

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-apti 7868  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-ltxr 7938  df-sub 8071  df-neg 8072

This theorem is referenced by: (None)