Theorem ltfintr 4460

Description: Transitivity law for finite less than. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
ltfintr	|- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> ((<<A, B>> e. <fin /\ <<B, C>> e. <fin ) -> <<A, C>> e. <fin ))

Proof of Theorem ltfintr

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		an4 797	. . 3 |- (((A =/= (/) /\ E.x e. Nn B = ((A +o x) +o 1c)) /\ (B =/= (/) /\ E.y e. Nn C = ((B +o y) +o 1c))) <-> ((A =/= (/) /\ B =/= (/)) /\ (E.x e. Nn B = ((A +o x) +o 1c) /\ E.y e. Nn C = ((B +o y) +o 1c))))
2		simpl 443	. . . . 5 |- ((A =/= (/) /\ B =/= (/)) -> A =/= (/))
3	2	a1i 10	. . . 4 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> ((A =/= (/) /\ B =/= (/)) -> A =/= (/)))
4		reeanv 2779	. . . . 5 |- (E.x e. Nn E.y e. Nn (B = ((A +o x) +o 1c) /\ C = ((B +o y) +o 1c)) <-> (E.x e. Nn B = ((A +o x) +o 1c) /\ E.y e. Nn C = ((B +o y) +o 1c)))
5		addccom 4407	. . . . . . . . . 10 |- (1c +o y) = (y +o 1c)
6		peano2 4404	. . . . . . . . . 10 |- (y e. Nn -> (y +o 1c) e. Nn )
7	5, 6	syl5eqel 2437	. . . . . . . . 9 |- (y e. Nn -> (1c +o y) e. Nn )
8		nncaddccl 4420	. . . . . . . . 9 |- ((x e. Nn /\ (1c +o y) e. Nn ) -> (x +o (1c +o y)) e. Nn )
9	7, 8	sylan2 460	. . . . . . . 8 |- ((x e. Nn /\ y e. Nn ) -> (x +o (1c +o y)) e. Nn )
10	9	adantl 452	. . . . . . 7 |- (((A e. Nn /\ B e. Nn /\ C e. Nn ) /\ (x e. Nn /\ y e. Nn )) -> (x +o (1c +o y)) e. Nn )
11		addceq1 4384	. . . . . . . . . 10 |- (B = ((A +o x) +o 1c) -> (B +o y) = (((A +o x) +o 1c) +o y))
12		addceq1 4384	. . . . . . . . . 10 |- ((B +o y) = (((A +o x) +o 1c) +o y) -> ((B +o y) +o 1c) = ((((A +o x) +o 1c) +o y) +o 1c))
13	11, 12	syl 15	. . . . . . . . 9 |- (B = ((A +o x) +o 1c) -> ((B +o y) +o 1c) = ((((A +o x) +o 1c) +o y) +o 1c))
14	13	eqeq2d 2364	. . . . . . . 8 |- (B = ((A +o x) +o 1c) -> (C = ((B +o y) +o 1c) <-> C = ((((A +o x) +o 1c) +o y) +o 1c)))
15	14	biimpa 470	. . . . . . 7 |- ((B = ((A +o x) +o 1c) /\ C = ((B +o y) +o 1c)) -> C = ((((A +o x) +o 1c) +o y) +o 1c))
16		addceq2 4385	. . . . . . . . . . . 12 |- (z = (x +o (1c +o y)) -> (A +o z) = (A +o (x +o (1c +o y))))
17		addcass 4416	. . . . . . . . . . . . 13 |- (((A +o x) +o 1c) +o y) = ((A +o x) +o (1c +o y))
18		addcass 4416	. . . . . . . . . . . . 13 |- ((A +o x) +o (1c +o y)) = (A +o (x +o (1c +o y)))
19	17, 18	eqtri 2373	. . . . . . . . . . . 12 |- (((A +o x) +o 1c) +o y) = (A +o (x +o (1c +o y)))
20	16, 19	syl6eqr 2403	. . . . . . . . . . 11 |- (z = (x +o (1c +o y)) -> (A +o z) = (((A +o x) +o 1c) +o y))
21		addceq1 4384	. . . . . . . . . . 11 |- ((A +o z) = (((A +o x) +o 1c) +o y) -> ((A +o z) +o 1c) = ((((A +o x) +o 1c) +o y) +o 1c))
22	20, 21	syl 15	. . . . . . . . . 10 |- (z = (x +o (1c +o y)) -> ((A +o z) +o 1c) = ((((A +o x) +o 1c) +o y) +o 1c))
23	22	eqeq2d 2364	. . . . . . . . 9 |- (z = (x +o (1c +o y)) -> (C = ((A +o z) +o 1c) <-> C = ((((A +o x) +o 1c) +o y) +o 1c)))
24	23	rspcev 2956	. . . . . . . 8 |- (((x +o (1c +o y)) e. Nn /\ C = ((((A +o x) +o 1c) +o y) +o 1c)) -> E.z e. Nn C = ((A +o z) +o 1c))
25	24	ex 423	. . . . . . 7 |- ((x +o (1c +o y)) e. Nn -> (C = ((((A +o x) +o 1c) +o y) +o 1c) -> E.z e. Nn C = ((A +o z) +o 1c)))
26	10, 15, 25	syl2im 34	. . . . . 6 |- (((A e. Nn /\ B e. Nn /\ C e. Nn ) /\ (x e. Nn /\ y e. Nn )) -> ((B = ((A +o x) +o 1c) /\ C = ((B +o y) +o 1c)) -> E.z e. Nn C = ((A +o z) +o 1c)))
27	26	rexlimdvva 2746	. . . . 5 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> (E.x e. Nn E.y e. Nn (B = ((A +o x) +o 1c) /\ C = ((B +o y) +o 1c)) -> E.z e. Nn C = ((A +o z) +o 1c)))
28	4, 27	syl5bir 209	. . . 4 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> ((E.x e. Nn B = ((A +o x) +o 1c) /\ E.y e. Nn C = ((B +o y) +o 1c)) -> E.z e. Nn C = ((A +o z) +o 1c)))
29	3, 28	anim12d 546	. . 3 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> (((A =/= (/) /\ B =/= (/)) /\ (E.x e. Nn B = ((A +o x) +o 1c) /\ E.y e. Nn C = ((B +o y) +o 1c))) -> (A =/= (/) /\ E.z e. Nn C = ((A +o z) +o 1c))))
30	1, 29	syl5bi 208	. 2 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> (((A =/= (/) /\ E.x e. Nn B = ((A +o x) +o 1c)) /\ (B =/= (/) /\ E.y e. Nn C = ((B +o y) +o 1c))) -> (A =/= (/) /\ E.z e. Nn C = ((A +o z) +o 1c))))
31		opkltfing 4450	. . . 4 |- ((A e. Nn /\ B e. Nn ) -> (<<A, B>> e. <fin <-> (A =/= (/) /\ E.x e. Nn B = ((A +o x) +o 1c))))
32	31	3adant3 975	. . 3 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> (<<A, B>> e. <fin <-> (A =/= (/) /\ E.x e. Nn B = ((A +o x) +o 1c))))
33		opkltfing 4450	. . . 4 |- ((B e. Nn /\ C e. Nn ) -> (<<B, C>> e. <fin <-> (B =/= (/) /\ E.y e. Nn C = ((B +o y) +o 1c))))
34	33	3adant1 973	. . 3 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> (<<B, C>> e. <fin <-> (B =/= (/) /\ E.y e. Nn C = ((B +o y) +o 1c))))
35	32, 34	anbi12d 691	. 2 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> ((<<A, B>> e. <fin /\ <<B, C>> e. <fin ) <-> ((A =/= (/) /\ E.x e. Nn B = ((A +o x) +o 1c)) /\ (B =/= (/) /\ E.y e. Nn C = ((B +o y) +o 1c)))))
36		opkltfing 4450	. . 3 |- ((A e. Nn /\ C e. Nn ) -> (<<A, C>> e. <fin <-> (A =/= (/) /\ E.z e. Nn C = ((A +o z) +o 1c))))
37	36	3adant2 974	. 2 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> (<<A, C>> e. <fin <-> (A =/= (/) /\ E.z e. Nn C = ((A +o z) +o 1c))))
38	30, 35, 37	3imtr4d 259	1 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> ((<<A, B>> e. <fin /\ <<B, C>> e. <fin ) -> <<A, C>> e. <fin ))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616  (/)c0 3551  <<copk 4058  1_cc1c 4135   Nn cnnc 4374   +o cplc 4376   <_fin cltfin 4434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-0c 4378  df-addc 4379  df-nnc 4380  df-ltfin 4442

This theorem is referenced by:  ltfinasym  4461  ltfintri  4467