Theorem ltordlem 10553

Description: Lemma for ltord1 10554. (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 2971	. 2 |- ( ph -> A. x e. S A. y e. S ( x < y -> A < B ) )
3		breq1 4656	. . . 4 |- ( x = C -> ( x < y <-> C < y ) )
4		ltord.2	. . . . 5 |- ( x = C -> A = M )
5	4	breq1d 4663	. . . 4 |- ( x = C -> ( A < B <-> M < B ) )
6	3, 5	imbi12d 334	. . 3 |- ( x = C -> ( ( x < y -> A < B ) <-> ( C < y -> M < B ) ) )
7		breq2 4657	. . . 4 |- ( y = D -> ( C < y <-> C < D ) )
8		eqeq1 2626	. . . . . . 7 |- ( x = y -> ( x = D <-> y = D ) )
9		ltord.1	. . . . . . . 8 |- ( x = y -> A = B )
10	9	eqeq1d 2624	. . . . . . 7 |- ( x = y -> ( A = N <-> B = N ) )
11	8, 10	imbi12d 334	. . . . . 6 |- ( x = y -> ( ( x = D -> A = N ) <-> ( y = D -> B = N ) ) )
12		ltord.3	. . . . . 6 |- ( x = D -> A = N )
13	11, 12	chvarv 2263	. . . . 5 |- ( y = D -> B = N )
14	13	breq2d 4665	. . . 4 |- ( y = D -> ( M < B <-> M < N ) )
15	7, 14	imbi12d 334	. . 3 |- ( y = D -> ( ( C < y -> M < B ) <-> ( C < D -> M < N ) ) )
16	6, 15	rspc2v 3322	. 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 486	1 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D -> M < N ) )

Syntax hints:   -> wi 4   /\ 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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  ltord1  10554