Theorem tsrlemax 17220

Description: Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)

Hypothesis

Ref	Expression
istsr.1	|- X = dom R

Assertion

Ref	Expression
tsrlemax	|- ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A R if ( B R C , C , B ) <-> ( A R B \/ A R C ) ) )

Proof of Theorem tsrlemax

Step	Hyp	Ref	Expression
1		breq2 4657	. . 3 |- ( C = if ( B R C , C , B ) -> ( A R C <-> A R if ( B R C , C , B ) ) )
2	1	bibi1d 333	. 2 |- ( C = if ( B R C , C , B ) -> ( ( A R C <-> ( A R B \/ A R C ) ) <-> ( A R if ( B R C , C , B ) <-> ( A R B \/ A R C ) ) ) )
3		breq2 4657	. . 3 |- ( B = if ( B R C , C , B ) -> ( A R B <-> A R if ( B R C , C , B ) ) )
4	3	bibi1d 333	. 2 |- ( B = if ( B R C , C , B ) -> ( ( A R B <-> ( A R B \/ A R C ) ) <-> ( A R if ( B R C , C , B ) <-> ( A R B \/ A R C ) ) ) )
5		olc 399	. . 3 |- ( A R C -> ( A R B \/ A R C ) )
6		eqid 2622	. . . . . . . . . 10 |- dom R = dom R
7	6	istsr 17217	. . . . . . . . 9 |- ( R e. TosetRel <-> ( R e. PosetRel /\ ( dom R X. dom R ) C_ ( R u. `' R ) ) )
8	7	simplbi 476	. . . . . . . 8 |- ( R e. TosetRel -> R e. PosetRel )
9		pstr 17211	. . . . . . . . 9 |- ( ( R e. PosetRel /\ A R B /\ B R C ) -> A R C )
10	9	3expib 1268	. . . . . . . 8 |- ( R e. PosetRel -> ( ( A R B /\ B R C ) -> A R C ) )
11	8, 10	syl 17	. . . . . . 7 |- ( R e. TosetRel -> ( ( A R B /\ B R C ) -> A R C ) )
12	11	adantr 481	. . . . . 6 |- ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A R B /\ B R C ) -> A R C ) )
13	12	expdimp 453	. . . . 5 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R B ) -> ( B R C -> A R C ) )
14	13	impancom 456	. . . 4 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ B R C ) -> ( A R B -> A R C ) )
15		idd 24	. . . 4 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ B R C ) -> ( A R C -> A R C ) )
16	14, 15	jaod 395	. . 3 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ B R C ) -> ( ( A R B \/ A R C ) -> A R C ) )
17	5, 16	impbid2 216	. 2 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ B R C ) -> ( A R C <-> ( A R B \/ A R C ) ) )
18		orc 400	. . 3 |- ( A R B -> ( A R B \/ A R C ) )
19		idd 24	. . . 4 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ -. B R C ) -> ( A R B -> A R B ) )
20		istsr.1	. . . . . . . 8 |- X = dom R
21	20	tsrlin 17219	. . . . . . 7 |- ( ( R e. TosetRel /\ B e. X /\ C e. X ) -> ( B R C \/ C R B ) )
22	21	3adant3r1 1274	. . . . . 6 |- ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B R C \/ C R B ) )
23	22	orcanai 952	. . . . 5 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ -. B R C ) -> C R B )
24		pstr 17211	. . . . . . . . . 10 |- ( ( R e. PosetRel /\ A R C /\ C R B ) -> A R B )
25	24	3expib 1268	. . . . . . . . 9 |- ( R e. PosetRel -> ( ( A R C /\ C R B ) -> A R B ) )
26	8, 25	syl 17	. . . . . . . 8 |- ( R e. TosetRel -> ( ( A R C /\ C R B ) -> A R B ) )
27	26	adantr 481	. . . . . . 7 |- ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A R C /\ C R B ) -> A R B ) )
28	27	expdimp 453	. . . . . 6 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ A R C ) -> ( C R B -> A R B ) )
29	28	impancom 456	. . . . 5 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ C R B ) -> ( A R C -> A R B ) )
30	23, 29	syldan 487	. . . 4 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ -. B R C ) -> ( A R C -> A R B ) )
31	19, 30	jaod 395	. . 3 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ -. B R C ) -> ( ( A R B \/ A R C ) -> A R B ) )
32	18, 31	impbid2 216	. 2 |- ( ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) /\ -. B R C ) -> ( A R B <-> ( A R B \/ A R C ) ) )
33	2, 4, 17, 32	ifbothda 4123	1 |- ( ( R e. TosetRel /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A R if ( B R C , C , B ) <-> ( A R B \/ A R C ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   ifcif 4086   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   PosetRelcps 17198   TosetRel ctsr 17199

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  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-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-ps 17200  df-tsr 17201

This theorem is referenced by:  ordtbaslem  20992