Theorem brecop2 7728

Description: Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.)

Hypotheses

Ref	Expression
brecop2.1	⊢ ∼ ∈ V
brecop2.5	⊢ dom ∼ = (𝐺 × 𝐺)
brecop2.6	⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ )
brecop2.7	⊢ 𝑅 ⊆ (𝐻 × 𝐻)
brecop2.8	⊢ ≤ ⊆ (𝐺 × 𝐺)
brecop2.9	⊢ ¬ ∅ ∈ 𝐺
brecop2.10	⊢ dom + = (𝐺 × 𝐺)
brecop2.11	⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([⟨𝐴, 𝐵⟩] ∼ 𝑅[⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) ≤ (𝐵 + 𝐶)))

Assertion

Ref	Expression
brecop2	⊢ ([⟨𝐴, 𝐵⟩] ∼ 𝑅[⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) ≤ (𝐵 + 𝐶))

Proof of Theorem brecop2

Step	Hyp	Ref	Expression
1		brecop2.7	. . . 4 ⊢ 𝑅 ⊆ (𝐻 × 𝐻)
2	1	brel 5090	. . 3 ⊢ ([⟨𝐴, 𝐵⟩] ∼ 𝑅[⟨𝐶, 𝐷⟩] ∼ → ([⟨𝐴, 𝐵⟩] ∼ ∈ 𝐻 ∧ [⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻))
3		brecop2.5	. . . . . . 7 ⊢ dom ∼ = (𝐺 × 𝐺)
4		ecelqsdm 7704	. . . . . . 7 ⊢ ((dom ∼ = (𝐺 × 𝐺) ∧ [⟨𝐴, 𝐵⟩] ∼ ∈ ((𝐺 × 𝐺) / ∼ )) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
5	3, 4	mpan 702	. . . . . 6 ⊢ ([⟨𝐴, 𝐵⟩] ∼ ∈ ((𝐺 × 𝐺) / ∼ ) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
6		brecop2.6	. . . . . 6 ⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ )
7	5, 6	eleq2s 2706	. . . . 5 ⊢ ([⟨𝐴, 𝐵⟩] ∼ ∈ 𝐻 → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
8		opelxp 5070	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺) ↔ (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺))
9	7, 8	sylib 207	. . . 4 ⊢ ([⟨𝐴, 𝐵⟩] ∼ ∈ 𝐻 → (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺))
10		ecelqsdm 7704	. . . . . . 7 ⊢ ((dom ∼ = (𝐺 × 𝐺) ∧ [⟨𝐶, 𝐷⟩] ∼ ∈ ((𝐺 × 𝐺) / ∼ )) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
11	3, 10	mpan 702	. . . . . 6 ⊢ ([⟨𝐶, 𝐷⟩] ∼ ∈ ((𝐺 × 𝐺) / ∼ ) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
12	11, 6	eleq2s 2706	. . . . 5 ⊢ ([⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻 → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
13		opelxp 5070	. . . . 5 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺) ↔ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))
14	12, 13	sylib 207	. . . 4 ⊢ ([⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻 → (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))
15	9, 14	anim12i 588	. . 3 ⊢ (([⟨𝐴, 𝐵⟩] ∼ ∈ 𝐻 ∧ [⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻) → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)))
16	2, 15	syl 17	. 2 ⊢ ([⟨𝐴, 𝐵⟩] ∼ 𝑅[⟨𝐶, 𝐷⟩] ∼ → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)))
17		brecop2.8	. . . . 5 ⊢ ≤ ⊆ (𝐺 × 𝐺)
18	17	brel 5090	. . . 4 ⊢ ((𝐴 + 𝐷) ≤ (𝐵 + 𝐶) → ((𝐴 + 𝐷) ∈ 𝐺 ∧ (𝐵 + 𝐶) ∈ 𝐺))
19		brecop2.10	. . . . . 6 ⊢ dom + = (𝐺 × 𝐺)
20		brecop2.9	. . . . . 6 ⊢ ¬ ∅ ∈ 𝐺
21	19, 20	ndmovrcl 6718	. . . . 5 ⊢ ((𝐴 + 𝐷) ∈ 𝐺 → (𝐴 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))
22	19, 20	ndmovrcl 6718	. . . . 5 ⊢ ((𝐵 + 𝐶) ∈ 𝐺 → (𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐺))
23	21, 22	anim12i 588	. . . 4 ⊢ (((𝐴 + 𝐷) ∈ 𝐺 ∧ (𝐵 + 𝐶) ∈ 𝐺) → ((𝐴 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) ∧ (𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐺)))
24	18, 23	syl 17	. . 3 ⊢ ((𝐴 + 𝐷) ≤ (𝐵 + 𝐶) → ((𝐴 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) ∧ (𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐺)))
25		an42 862	. . 3 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) ∧ (𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐺)) ↔ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)))
26	24, 25	sylib 207	. 2 ⊢ ((𝐴 + 𝐷) ≤ (𝐵 + 𝐶) → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)))
27		brecop2.11	. 2 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([⟨𝐴, 𝐵⟩] ∼ 𝑅[⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) ≤ (𝐵 + 𝐶)))
28	16, 26, 27	pm5.21nii 367	1 ⊢ ([⟨𝐴, 𝐵⟩] ∼ 𝑅[⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) ≤ (𝐵 + 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038  (class class class)co 6549  [cec 7627   / cqs 7628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ov 6552  df-ec 7631  df-qs 7635

This theorem is referenced by:  ltsrpr  9777