Theorem tsrlemax 18653

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 5155	. . 3 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐶 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵)))
2	1	bibi1d 343	. 2 ⊢ (𝐶 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))))
3		breq2 5155	. . 3 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → (𝐴𝑅𝐵 ↔ 𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵)))
4	3	bibi1d 343	. 2 ⊢ (𝐵 = if(𝐵𝑅𝐶, 𝐶, 𝐵) → ((𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)) ↔ (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))))
5		olc 869	. . 3 ⊢ (𝐴𝑅𝐶 → (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))
6		eqid 2737	. . . . . . . . . 10 ⊢ dom 𝑅 = dom 𝑅
7	6	istsr 18650	. . . . . . . . 9 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅)))
8	7	simplbi 497	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
9		pstr 18644	. . . . . . . . 9 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
10	9	3expib 1123	. . . . . . . 8 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
11	8, 10	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ TosetRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
12	11	adantr 480	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
13	12	expdimp 452	. . . . 5 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅𝐵) → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
14	13	impancom 451	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
15		idd 24	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 → 𝐴𝑅𝐶))
16	14, 15	jaod 860	. . 3 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶) → 𝐴𝑅𝐶))
17	5, 16	impbid2 226	. 2 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))
18		orc 868	. . 3 ⊢ (𝐴𝑅𝐵 → (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶))
19		idd 24	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐵))
20		istsr.1	. . . . . . . 8 ⊢ 𝑋 = dom 𝑅
21	20	tsrlin 18652	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))
22	21	3adant3r1 1183	. . . . . 6 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))
23	22	orcanai 1005	. . . . 5 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → 𝐶𝑅𝐵)
24		pstr 18644	. . . . . . . . . 10 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵)
25	24	3expib 1123	. . . . . . . . 9 ⊢ (𝑅 ∈ PosetRel → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵))
26	8, 25	syl 17	. . . . . . . 8 ⊢ (𝑅 ∈ TosetRel → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵))
27	26	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐴𝑅𝐵))
28	27	expdimp 452	. . . . . 6 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴𝑅𝐶) → (𝐶𝑅𝐵 → 𝐴𝑅𝐵))
29	28	impancom 451	. . . . 5 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶𝑅𝐵) → (𝐴𝑅𝐶 → 𝐴𝑅𝐵))
30	23, 29	syldan 591	. . . 4 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐶 → 𝐴𝑅𝐵))
31	19, 30	jaod 860	. . 3 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → ((𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶) → 𝐴𝑅𝐵))
32	18, 31	impbid2 226	. 2 ⊢ (((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ ¬ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))
33	2, 4, 17, 32	ifbothda 4572	1 ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1539   ∈ wcel 2108   ∪ cun 3964   ⊆ wss 3966  ifcif 4534   class class class wbr 5151   × cxp 5691  ^◡ccnv 5692  dom cdm 5693  PosetRelcps 18631   TosetRel ctsr 18632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-res 5705  df-ps 18633  df-tsr 18634

This theorem is referenced by:  ordtbaslem  23221