Theorem xporderlem 7815

Description: Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)

Hypothesis

Ref	Expression
xporderlem.1	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}

Assertion

Ref	Expression
xporderlem	⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑎,𝑦   𝑥,𝑏,𝑦   𝑥,𝑐,𝑦   𝑥,𝑑,𝑦

Allowed substitution hints:   𝐴(𝑎,𝑏,𝑐,𝑑)   𝐵(𝑎,𝑏,𝑐,𝑑)   𝑅(𝑎,𝑏,𝑐,𝑑)   𝑆(𝑎,𝑏,𝑐,𝑑)   𝑇(𝑥,𝑦,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem xporderlem

Step	Hyp	Ref	Expression
1		df-br 5060	. . 3 ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ 𝑇)
2		xporderlem.1	. . . 4 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}
3	2	eleq2i 2904	. . 3 ⊢ (⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ 𝑇 ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))})
4	1, 3	bitri 277	. 2 ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))})
5		opex 5349	. . 3 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
6		opex 5349	. . 3 ⊢ ⟨𝑐, 𝑑⟩ ∈ V
7		eleq1 2900	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵)))
8		opelxp 5586	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵))
9	7, 8	syl6bb 289	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 ∈ (𝐴 × 𝐵) ↔ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)))
10	9	anbi1d 631	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵))))
11		vex 3498	. . . . . . 7 ⊢ 𝑎 ∈ V
12		vex 3498	. . . . . . 7 ⊢ 𝑏 ∈ V
13	11, 12	op1std 7693	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (1st ‘𝑥) = 𝑎)
14	13	breq1d 5069	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st ‘𝑥)𝑅(1st ‘𝑦) ↔ 𝑎𝑅(1st ‘𝑦)))
15	13	eqeq1d 2823	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((1st ‘𝑥) = (1st ‘𝑦) ↔ 𝑎 = (1st ‘𝑦)))
16	11, 12	op2ndd 7694	. . . . . . 7 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (2nd ‘𝑥) = 𝑏)
17	16	breq1d 5069	. . . . . 6 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ↔ 𝑏𝑆(2nd ‘𝑦)))
18	15, 17	anbi12d 632	. . . . 5 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦)) ↔ (𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦))))
19	14, 18	orbi12d 915	. . . 4 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))) ↔ (𝑎𝑅(1st ‘𝑦) ∨ (𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦)))))
20	10, 19	anbi12d 632	. . 3 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦)))) ↔ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ (𝑎𝑅(1st ‘𝑦) ∨ (𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦))))))
21		eleq1 2900	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ ⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵)))
22		opelxp 5586	. . . . . 6 ⊢ (⟨𝑐, 𝑑⟩ ∈ (𝐴 × 𝐵) ↔ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵))
23	21, 22	syl6bb 289	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑦 ∈ (𝐴 × 𝐵) ↔ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)))
24	23	anbi2d 630	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵))))
25		vex 3498	. . . . . . 7 ⊢ 𝑐 ∈ V
26		vex 3498	. . . . . . 7 ⊢ 𝑑 ∈ V
27	25, 26	op1std 7693	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (1st ‘𝑦) = 𝑐)
28	27	breq2d 5071	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎𝑅(1st ‘𝑦) ↔ 𝑎𝑅𝑐))
29	27	eqeq2d 2832	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑎 = (1st ‘𝑦) ↔ 𝑎 = 𝑐))
30	25, 26	op2ndd 7694	. . . . . . 7 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (2nd ‘𝑦) = 𝑑)
31	30	breq2d 5071	. . . . . 6 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → (𝑏𝑆(2nd ‘𝑦) ↔ 𝑏𝑆𝑑))
32	29, 31	anbi12d 632	. . . . 5 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦)) ↔ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑)))
33	28, 32	orbi12d 915	. . . 4 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → ((𝑎𝑅(1st ‘𝑦) ∨ (𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦))) ↔ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))
34	24, 33	anbi12d 632	. . 3 ⊢ (𝑦 = ⟨𝑐, 𝑑⟩ → ((((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ (𝑎𝑅(1st ‘𝑦) ∨ (𝑎 = (1st ‘𝑦) ∧ 𝑏𝑆(2nd ‘𝑦)))) ↔ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑)))))
35	5, 6, 20, 34	opelopab 5422	. 2 ⊢ (⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))} ↔ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))
36		an4 654	. . 3 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)))
37	36	anbi1i 625	. 2 ⊢ ((((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))) ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))
38	4, 35, 37	3bitri 299	1 ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110  ⟨cop 4567   class class class wbr 5059  {copab 5121   × cxp 5548  ‘cfv 6350  1^st c1st 7681  2^nd c2nd 7682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fv 6358  df-1st 7683  df-2nd 7684

This theorem is referenced by:  poxp  7816  soxp  7817