Theorem ltordlem 8352

Description: Lemma for eqord1 8353. (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
ltordlem	|- ( ( 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 ltordlem

Step	Hyp	Ref	Expression
1		ltord.6	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) )
2	1	ralrimivva 2539	. 2 |- ( ph -> A. x e. S A. y e. S ( x < y -> A < B ) )
3		breq1 3968	. . . 4 |- ( x = C -> ( x < y <-> C < y ) )
4		ltord.2	. . . . 5 |- ( x = C -> A = M )
5	4	breq1d 3975	. . . 4 |- ( x = C -> ( A < B <-> M < B ) )
6	3, 5	imbi12d 233	. . 3 |- ( x = C -> ( ( x < y -> A < B ) <-> ( C < y -> M < B ) ) )
7		breq2 3969	. . . 4 |- ( y = D -> ( C < y <-> C < D ) )
8		eqeq1 2164	. . . . . . 7 |- ( x = y -> ( x = D <-> y = D ) )
9		ltord.1	. . . . . . . 8 |- ( x = y -> A = B )
10	9	eqeq1d 2166	. . . . . . 7 |- ( x = y -> ( A = N <-> B = N ) )
11	8, 10	imbi12d 233	. . . . . 6 |- ( x = y -> ( ( x = D -> A = N ) <-> ( y = D -> B = N ) ) )
12		ltord.3	. . . . . 6 |- ( x = D -> A = N )
13	11, 12	chvarv 1917	. . . . 5 |- ( y = D -> B = N )
14	13	breq2d 3977	. . . 4 |- ( y = D -> ( M < B <-> M < N ) )
15	7, 14	imbi12d 233	. . 3 |- ( y = D -> ( ( C < y -> M < B ) <-> ( C < D -> M < N ) ) )
16	6, 15	rspc2v 2829	. 2 |- ( ( C e. S /\ D e. S ) -> ( A. x e. S A. y e. S ( x < y -> A < B ) -> ( C < D -> M < N ) ) )
17	2, 16	mpan9 279	1 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D -> M < N ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   A.wral 2435   C_ wss 3102   class class class wbr 3965   RRcr 7726   < clt 7907

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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966

This theorem is referenced by:  eqord1  8353