Theorem ltord1 10554

Description: Infer an ordering relation from a proof in only one direction. (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 )
ltord.6	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) )

Assertion

Ref	Expression
ltord1	|- ( ( 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 ltord1

Step	Hyp	Ref	Expression
1		ltord.1	. . 3 |- ( x = y -> A = B )
2		ltord.2	. . 3 |- ( x = C -> A = M )
3		ltord.3	. . 3 |- ( x = D -> A = N )
4		ltord.4	. . 3 |- S C_ RR
5		ltord.5	. . 3 |- ( ( ph /\ x e. S ) -> A e. RR )
6		ltord.6	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) )
7	1, 2, 3, 4, 5, 6	ltordlem 10553	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D -> M < N ) )
8		eqeq1 2626	. . . . . . . 8 |- ( x = C -> ( x = D <-> C = D ) )
9	2	eqeq1d 2624	. . . . . . . 8 |- ( x = C -> ( A = N <-> M = N ) )
10	8, 9	imbi12d 334	. . . . . . 7 |- ( x = C -> ( ( x = D -> A = N ) <-> ( C = D -> M = N ) ) )
11	10, 3	vtoclg 3266	. . . . . 6 |- ( C e. S -> ( C = D -> M = N ) )
12	11	ad2antrl 764	. . . . 5 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D -> M = N ) )
13	1, 3, 2, 4, 5, 6	ltordlem 10553	. . . . . 6 |- ( ( ph /\ ( D e. S /\ C e. S ) ) -> ( D < C -> N < M ) )
14	13	ancom2s 844	. . . . 5 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( D < C -> N < M ) )
15	12, 14	orim12d 883	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( ( C = D \/ D < C ) -> ( M = N \/ N < M ) ) )
16	15	con3d 148	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( -. ( M = N \/ N < M ) -> -. ( C = D \/ D < C ) ) )
17	5	ralrimiva 2966	. . . . . 6 |- ( ph -> A. x e. S A e. RR )
18	2	eleq1d 2686	. . . . . . 7 |- ( x = C -> ( A e. RR <-> M e. RR ) )
19	18	rspccva 3308	. . . . . 6 |- ( ( A. x e. S A e. RR /\ C e. S ) -> M e. RR )
20	17, 19	sylan 488	. . . . 5 |- ( ( ph /\ C e. S ) -> M e. RR )
21	3	eleq1d 2686	. . . . . . 7 |- ( x = D -> ( A e. RR <-> N e. RR ) )
22	21	rspccva 3308	. . . . . 6 |- ( ( A. x e. S A e. RR /\ D e. S ) -> N e. RR )
23	17, 22	sylan 488	. . . . 5 |- ( ( ph /\ D e. S ) -> N e. RR )
24	20, 23	anim12dan 882	. . . 4 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M e. RR /\ N e. RR ) )
25		axlttri 10109	. . . 4 |- ( ( M e. RR /\ N e. RR ) -> ( M < N <-> -. ( M = N \/ N < M ) ) )
26	24, 25	syl 17	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M < N <-> -. ( M = N \/ N < M ) ) )
27	4	sseli 3599	. . . . 5 |- ( C e. S -> C e. RR )
28	4	sseli 3599	. . . . 5 |- ( D e. S -> D e. RR )
29		axlttri 10109	. . . . 5 |- ( ( C e. RR /\ D e. RR ) -> ( C < D <-> -. ( C = D \/ D < C ) ) )
30	27, 28, 29	syl2an 494	. . . 4 |- ( ( C e. S /\ D e. S ) -> ( C < D <-> -. ( C = D \/ D < C ) ) )
31	30	adantl 482	. . 3 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D <-> -. ( C = D \/ D < C ) ) )
32	16, 26, 31	3imtr4d 283	. 2 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( M < N -> C < D ) )
33	7, 32	impbid 202	1 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D <-> M < N ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653   RRcr 9935   < clt 10074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-pre-lttri 10010

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079

This theorem is referenced by:  leord1  10555  ltord2  10557  ltexp2  12914  eflt  14847  tanord1  24283  tanord  24284  monotuz  37506  monotoddzzfi  37507  rpexpmord  37513