Theorem tsrlemax 18641

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 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))

Proof of Theorem tsrlemax

Step	Hyp	Ref	Expression
1		breq2 5112	. . 3 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐶 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵)))
2	1	bibi1d 346	. 2 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))))
3		breq2 5112	. . 3 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐵 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵)))
4	3	bibi1d 346	. 2 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))))
5		olc 881	. . 3 ⊢ (𝐴𝑅𝐶 → (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))
6		eqid 2761	. . . . . . . . . 10 ⊢ dom 𝑅 = dom 𝑅
7	6	istsr 18638	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅)))
8	7	simplbi 501	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
9		pstr 18632	. . . . . . . . 9 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
10	9	3expib 1138	. . . . . . . 8 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
11	8, 10	syl 18	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
12	11	adantr 485	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
13	12	expdimp 457	. . . . 5 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
14	13	impancom 456	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
15		idd 25	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 → 𝐴𝑅𝐶))
16	14, 15	jaod 872	. . 3 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶) → 𝐴𝑅𝐶))
17	5, 16	impbid2 229	. 2 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))
18		orc 880	. . 3 ⊢ (𝐴𝑅𝐵 → (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))
19		idd 25	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐵))
20		istsr.1	. . . . . . . 8 ⊢ 𝑋 = dom 𝑅
21	20	tsrlin 18640	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))
22	21	3adant3r1 1199	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))
23	22	orcanai 1018	. . . . 5 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → 𝐶𝑅𝐵)
24		pstr 18632	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵)
25	24	3expib 1138	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵))
26	8, 25	syl 18	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵))
27	26	adantr 485	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵))
28	27	expdimp 457	. . . . . 6 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅𝐶) → (𝐶𝑅𝐵 → 𝐴𝑅𝐵))
29	28	impancom 456	. . . . 5 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶𝑅𝐵) → (𝐴𝑅𝐶 → 𝐴𝑅𝐵))
30	23, 29	syldan 602	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 → 𝐴𝑅𝐵))
31	19, 30	jaod 872	. . 3 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶) → 𝐴𝑅𝐵))
32	18, 31	impbid2 229	. 2 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))
33	2, 4, 17, 32	ifbothda 4525	1 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ∪ cun 3902   ⊆ wss 3904  ifcif 4486   class class class wbr 5108   × cxp 5659  ^◡ccnv 5660  dom cdm 5661  PosetRelcps 18619   TosetRel ctsr 18620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-ps 18621  df-tsr 18622

This theorem is referenced by:  ordtbaslem  23324