Theorem ltordlem 8590

Description: Lemma for eqord1 8591. (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 2590	. 2 |- ( ph -> A. x e. S A. y e. S ( x < y -> A < B ) )
3		breq1 4062	. . . 4 |- ( x = C -> ( x < y <-> C < y ) )
4		ltord.2	. . . . 5 |- ( x = C -> A = M )
5	4	breq1d 4069	. . . 4 |- ( x = C -> ( A < B <-> M < B ) )
6	3, 5	imbi12d 234	. . 3 |- ( x = C -> ( ( x < y -> A < B ) <-> ( C < y -> M < B ) ) )
7		breq2 4063	. . . 4 |- ( y = D -> ( C < y <-> C < D ) )
8		eqeq1 2214	. . . . . . 7 |- ( x = y -> ( x = D <-> y = D ) )
9		ltord.1	. . . . . . . 8 |- ( x = y -> A = B )
10	9	eqeq1d 2216	. . . . . . 7 |- ( x = y -> ( A = N <-> B = N ) )
11	8, 10	imbi12d 234	. . . . . 6 |- ( x = y -> ( ( x = D -> A = N ) <-> ( y = D -> B = N ) ) )
12		ltord.3	. . . . . 6 |- ( x = D -> A = N )
13	11, 12	chvarv 1966	. . . . 5 |- ( y = D -> B = N )
14	13	breq2d 4071	. . . 4 |- ( y = D -> ( M < B <-> M < N ) )
15	7, 14	imbi12d 234	. . 3 |- ( y = D -> ( ( C < y -> M < B ) <-> ( C < D -> M < N ) ) )
16	6, 15	rspc2v 2897	. 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 281	1 |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C < D -> M < N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   A.wral 2486   C_ wss 3174   class class class wbr 4059   RRcr 7959   < clt 8142

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060

This theorem is referenced by:  eqord1  8591