Theorem eqord2 8528

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 8238	. . 3 |- ( x = y -> -u A = -u B )
3		ltord.2	. . . 4 |- ( x = C -> A = M )
4	3	negeqd 8238	. . 3 |- ( x = C -> -u A = -u M )
5		ltord.3	. . . 4 |- ( x = D -> A = N )
6	5	negeqd 8238	. . 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 8423	. . 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 2570	. . . . . . 7 |- ( ph -> A. x e. S A e. RR )
12	1	eleq1d 2265	. . . . . . . 8 |- ( x = y -> ( A e. RR <-> B e. RR ) )
13	12	rspccva 2867	. . . . . . 7 |- ( ( A. x e. S A e. RR /\ y e. S ) -> B e. RR )
14	11, 13	sylan 283	. . . . . 6 |- ( ( ph /\ y e. S ) -> B e. RR )
15	14	adantrl 478	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> B e. RR )
16	8	adantrr 479	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> A e. RR )
17		ltneg 8506	. . . . 5 |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> -u A < -u B ) )
18	15, 16, 17	syl2anc 411	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( B < A <-> -u A < -u B ) )
19	10, 18	sylibd 149	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> -u A < -u B ) )
20	2, 4, 6, 7, 9, 19	eqord1 8527	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> -u M = -u N ) )
21	3	eleq1d 2265	. . . . . . 7 |- ( x = C -> ( A e. RR <-> M e. RR ) )
22	21	rspccva 2867	. . . . . 6 |- ( ( A. x e. S A e. RR /\ C e. S ) -> M e. RR )
23	11, 22	sylan 283	. . . . 5 |- ( ( ph /\ C e. S ) -> M e. RR )
24	23	adantrr 479	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. RR )
25	24	recnd 8072	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. CC )
26	5	eleq1d 2265	. . . . . . 7 |- ( x = D -> ( A e. RR <-> N e. RR ) )
27	26	rspccva 2867	. . . . . 6 |- ( ( A. x e. S A e. RR /\ D e. S ) -> N e. RR )
28	11, 27	sylan 283	. . . . 5 |- ( ( ph /\ D e. S ) -> N e. RR )
29	28	adantrl 478	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. RR )
30	29	recnd 8072	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. CC )
31	25, 30	neg11ad 8350	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( -u M = -u N <-> M = N ) )
32	20, 31	bitrd 188	1 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   class class class wbr 4034   RRcr 7895   < clt 8078   -ucneg 8215

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-apti 8011  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-sub 8216  df-neg 8217

This theorem is referenced by: (None)