Theorem eqord2 8443

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 8154	. . 3 |- ( x = y -> -u A = -u B )
3		ltord.2	. . . 4 |- ( x = C -> A = M )
4	3	negeqd 8154	. . 3 |- ( x = C -> -u A = -u M )
5		ltord.3	. . . 4 |- ( x = D -> A = N )
6	5	negeqd 8154	. . 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 8339	. . 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 2550	. . . . . . 7 |- ( ph -> A. x e. S A e. RR )
12	1	eleq1d 2246	. . . . . . . 8 |- ( x = y -> ( A e. RR <-> B e. RR ) )
13	12	rspccva 2842	. . . . . . 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 8421	. . . . 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 8442	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> -u M = -u N ) )
21	3	eleq1d 2246	. . . . . . 7 |- ( x = C -> ( A e. RR <-> M e. RR ) )
22	21	rspccva 2842	. . . . . 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 7988	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. CC )
26	5	eleq1d 2246	. . . . . . 7 |- ( x = D -> ( A e. RR <-> N e. RR ) )
27	26	rspccva 2842	. . . . . 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 7988	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. CC )
31	25, 30	neg11ad 8266	. 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 1353   e. wcel 2148   A.wral 2455   C_ wss 3131   class class class wbr 4005   RRcr 7812   < clt 7994   -ucneg 8131

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-apti 7928  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-sub 8132  df-neg 8133

This theorem is referenced by: (None)