Theorem eqord2 8760

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 8470	. . 3 |- ( x = y -> -u A = -u B )
3		ltord.2	. . . 4 |- ( x = C -> A = M )
4	3	negeqd 8470	. . 3 |- ( x = C -> -u A = -u M )
5		ltord.3	. . . 4 |- ( x = D -> A = N )
6	5	negeqd 8470	. . 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 8655	. . 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 2617	. . . . . . 7 |- ( ph -> A. x e. S A e. RR )
12	1	eleq1d 2303	. . . . . . . 8 |- ( x = y -> ( A e. RR <-> B e. RR ) )
13	12	rspccva 2922	. . . . . . 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 8738	. . . . 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 8759	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> -u M = -u N ) )
21	3	eleq1d 2303	. . . . . . 7 |- ( x = C -> ( A e. RR <-> M e. RR ) )
22	21	rspccva 2922	. . . . . 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 8304	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> M e. CC )
26	5	eleq1d 2303	. . . . . . 7 |- ( x = D -> ( A e. RR <-> N e. RR ) )
27	26	rspccva 2922	. . . . . 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 8304	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> N e. CC )
31	25, 30	neg11ad 8582	. 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 1398   e. wcel 2205   A.wral 2522   C_ wss 3213   class class class wbr 4111   RRcr 8128   < clt 8310   -ucneg 8447

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-sub 8448  df-neg 8449

This theorem is referenced by: (None)