Theorem leltfintr 4459

Description: Transitivity law for finite less than and less than or equal. (Contributed by SF, 2-Feb-2015.)

Assertion

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

Proof of Theorem leltfintr

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

Step	Hyp	Ref	Expression
1		opklefing 4449	. . . 4 |- ((A e. Nn /\ B e. Nn ) -> (<<A, B>> e. <_fin <-> E.x e. Nn B = (A +o x)))
2	1	3adant3 975	. . 3 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> (<<A, B>> e. <_fin <-> E.x e. Nn B = (A +o x)))
3		addcnnul 4454	. . . . . . . . . 10 |- ((A +o x) =/= (/) -> (A =/= (/) /\ x =/= (/)))
4	3	simpld 445	. . . . . . . . 9 |- ((A +o x) =/= (/) -> A =/= (/))
5	4	a1i 10	. . . . . . . 8 |- (((A e. Nn /\ C e. Nn ) /\ x e. Nn ) -> ((A +o x) =/= (/) -> A =/= (/)))
6		nncaddccl 4420	. . . . . . . . . . . . . 14 |- ((x e. Nn /\ y e. Nn ) -> (x +o y) e. Nn )
7	6	3adant1 973	. . . . . . . . . . . . 13 |- ((A e. Nn /\ x e. Nn /\ y e. Nn ) -> (x +o y) e. Nn )
8		addcass 4416	. . . . . . . . . . . . . . 15 |- ((A +o x) +o y) = (A +o (x +o y))
9		addceq1 4384	. . . . . . . . . . . . . . 15 |- (((A +o x) +o y) = (A +o (x +o y)) -> (((A +o x) +o y) +o 1c) = ((A +o (x +o y)) +o 1c))
10	8, 9	ax-mp 5	. . . . . . . . . . . . . 14 |- (((A +o x) +o y) +o 1c) = ((A +o (x +o y)) +o 1c)
11	10	a1i 10	. . . . . . . . . . . . 13 |- ((A e. Nn /\ x e. Nn /\ y e. Nn ) -> (((A +o x) +o y) +o 1c) = ((A +o (x +o y)) +o 1c))
12		addceq2 4385	. . . . . . . . . . . . . . . 16 |- (z = (x +o y) -> (A +o z) = (A +o (x +o y)))
13		addceq1 4384	. . . . . . . . . . . . . . . 16 |- ((A +o z) = (A +o (x +o y)) -> ((A +o z) +o 1c) = ((A +o (x +o y)) +o 1c))
14	12, 13	syl 15	. . . . . . . . . . . . . . 15 |- (z = (x +o y) -> ((A +o z) +o 1c) = ((A +o (x +o y)) +o 1c))
15	14	eqeq2d 2364	. . . . . . . . . . . . . 14 |- (z = (x +o y) -> ((((A +o x) +o y) +o 1c) = ((A +o z) +o 1c) <-> (((A +o x) +o y) +o 1c) = ((A +o (x +o y)) +o 1c)))
16	15	rspcev 2956	. . . . . . . . . . . . 13 |- (((x +o y) e. Nn /\ (((A +o x) +o y) +o 1c) = ((A +o (x +o y)) +o 1c)) -> E.z e. Nn (((A +o x) +o y) +o 1c) = ((A +o z) +o 1c))
17	7, 11, 16	syl2anc 642	. . . . . . . . . . . 12 |- ((A e. Nn /\ x e. Nn /\ y e. Nn ) -> E.z e. Nn (((A +o x) +o y) +o 1c) = ((A +o z) +o 1c))
18		eqeq1 2359	. . . . . . . . . . . . 13 |- (C = (((A +o x) +o y) +o 1c) -> (C = ((A +o z) +o 1c) <-> (((A +o x) +o y) +o 1c) = ((A +o z) +o 1c)))
19	18	rexbidv 2636	. . . . . . . . . . . 12 |- (C = (((A +o x) +o y) +o 1c) -> (E.z e. Nn C = ((A +o z) +o 1c) <-> E.z e. Nn (((A +o x) +o y) +o 1c) = ((A +o z) +o 1c)))
20	17, 19	syl5ibrcom 213	. . . . . . . . . . 11 |- ((A e. Nn /\ x e. Nn /\ y e. Nn ) -> (C = (((A +o x) +o y) +o 1c) -> E.z e. Nn C = ((A +o z) +o 1c)))
21	20	3expa 1151	. . . . . . . . . 10 |- (((A e. Nn /\ x e. Nn ) /\ y e. Nn ) -> (C = (((A +o x) +o y) +o 1c) -> E.z e. Nn C = ((A +o z) +o 1c)))
22	21	adantllr 699	. . . . . . . . 9 |- ((((A e. Nn /\ C e. Nn ) /\ x e. Nn ) /\ y e. Nn ) -> (C = (((A +o x) +o y) +o 1c) -> E.z e. Nn C = ((A +o z) +o 1c)))
23	22	rexlimdva 2739	. . . . . . . 8 |- (((A e. Nn /\ C e. Nn ) /\ x e. Nn ) -> (E.y e. Nn C = (((A +o x) +o y) +o 1c) -> E.z e. Nn C = ((A +o z) +o 1c)))
24	5, 23	anim12d 546	. . . . . . 7 |- (((A e. Nn /\ C e. Nn ) /\ x e. Nn ) -> (((A +o x) =/= (/) /\ E.y e. Nn C = (((A +o x) +o y) +o 1c)) -> (A =/= (/) /\ E.z e. Nn C = ((A +o z) +o 1c))))
25		addcexg 4394	. . . . . . . . 9 |- ((A e. Nn /\ x e. Nn ) -> (A +o x) e. _V)
26	25	adantlr 695	. . . . . . . 8 |- (((A e. Nn /\ C e. Nn ) /\ x e. Nn ) -> (A +o x) e. _V)
27		simplr 731	. . . . . . . 8 |- (((A e. Nn /\ C e. Nn ) /\ x e. Nn ) -> C e. Nn )
28		opkltfing 4450	. . . . . . . 8 |- (((A +o x) e. _V /\ C e. Nn ) -> (<<(A +o x), C>> e. <fin <-> ((A +o x) =/= (/) /\ E.y e. Nn C = (((A +o x) +o y) +o 1c))))
29	26, 27, 28	syl2anc 642	. . . . . . 7 |- (((A e. Nn /\ C e. Nn ) /\ x e. Nn ) -> (<<(A +o x), C>> e. <fin <-> ((A +o x) =/= (/) /\ E.y e. Nn C = (((A +o x) +o y) +o 1c))))
30		opkltfing 4450	. . . . . . . 8 |- ((A e. Nn /\ C e. Nn ) -> (<<A, C>> e. <fin <-> (A =/= (/) /\ E.z e. Nn C = ((A +o z) +o 1c))))
31	30	adantr 451	. . . . . . 7 |- (((A e. Nn /\ C e. Nn ) /\ x e. Nn ) -> (<<A, C>> e. <fin <-> (A =/= (/) /\ E.z e. Nn C = ((A +o z) +o 1c))))
32	24, 29, 31	3imtr4d 259	. . . . . 6 |- (((A e. Nn /\ C e. Nn ) /\ x e. Nn ) -> (<<(A +o x), C>> e. <fin -> <<A, C>> e. <fin ))
33		opkeq1 4060	. . . . . . . 8 |- (B = (A +o x) -> <<B, C>> = <<(A +o x), C>>)
34	33	eleq1d 2419	. . . . . . 7 |- (B = (A +o x) -> (<<B, C>> e. <fin <-> <<(A +o x), C>> e. <fin ))
35	34	imbi1d 308	. . . . . 6 |- (B = (A +o x) -> ((<<B, C>> e. <fin -> <<A, C>> e. <fin ) <-> (<<(A +o x), C>> e. <fin -> <<A, C>> e. <fin )))
36	32, 35	syl5ibrcom 213	. . . . 5 |- (((A e. Nn /\ C e. Nn ) /\ x e. Nn ) -> (B = (A +o x) -> (<<B, C>> e. <fin -> <<A, C>> e. <fin )))
37	36	rexlimdva 2739	. . . 4 |- ((A e. Nn /\ C e. Nn ) -> (E.x e. Nn B = (A +o x) -> (<<B, C>> e. <fin -> <<A, C>> e. <fin )))
38	37	3adant2 974	. . 3 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> (E.x e. Nn B = (A +o x) -> (<<B, C>> e. <fin -> <<A, C>> e. <fin )))
39	2, 38	sylbid 206	. 2 |- ((A e. Nn /\ B e. Nn /\ C e. Nn ) -> (<<A, B>> e. <_fin -> (<<B, C>> e. <fin -> <<A, C>> e. <fin )))
40	39	imp3a 420	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  _Vcvv 2860  (/)c0 3551  <<copk 4058  1_cc1c 4135   Nn cnnc 4374   +o cplc 4376   <__fin clefin 4433   <_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-lefin 4441  df-ltfin 4442

This theorem is referenced by:  lenltfin  4470  vfinncvntnn  4549