Theorem brecop2 8869

Description: Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.) (Revised by AV, 12-Jul-2022.)

Hypotheses

Ref	Expression
brecop2.1	⊢ dom ∼ = (𝐺 × 𝐺)
brecop2.2	⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ )
brecop2.3	⊢ 𝑅 ⊆ (𝐻 × 𝐻)
brecop2.4	⊢ ≤ ⊆ (𝐺 × 𝐺)
brecop2.5	⊢ ¬ ∅ ∈ 𝐺
brecop2.6	⊢ dom + = (𝐺 × 𝐺)
brecop2.7	⊢ (((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		brecop2.3	. . . 4 ⊢ 𝑅 ⊆ (𝐻 × 𝐻)
2	1	brel 5765	. . 3 ⊢ ([⟨𝐴, 𝐵⟩] ∼ 𝑅[⟨𝐶, 𝐷⟩] ∼ → ([⟨𝐴, 𝐵⟩] ∼ ∈ 𝐻 ∧ [⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻))
3		brecop2.1	. . . . . . 7 ⊢ dom ∼ = (𝐺 × 𝐺)
4		ecelqsdm 8845	. . . . . . 7 ⊢ ((dom ∼ = (𝐺 × 𝐺) ∧ [⟨𝐴, 𝐵⟩] ∼ ∈ ((𝐺 × 𝐺) / ∼ )) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
5	3, 4	mpan 689	. . . . . 6 ⊢ ([⟨𝐴, 𝐵⟩] ∼ ∈ ((𝐺 × 𝐺) / ∼ ) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
6		brecop2.2	. . . . . 6 ⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ )
7	5, 6	eleq2s 2862	. . . . 5 ⊢ ([⟨𝐴, 𝐵⟩] ∼ ∈ 𝐻 → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
8		opelxp 5736	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺) ↔ (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺))
9	7, 8	sylib 218	. . . 4 ⊢ ([⟨𝐴, 𝐵⟩] ∼ ∈ 𝐻 → (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺))
10		ecelqsdm 8845	. . . . . . 7 ⊢ ((dom ∼ = (𝐺 × 𝐺) ∧ [⟨𝐶, 𝐷⟩] ∼ ∈ ((𝐺 × 𝐺) / ∼ )) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
11	3, 10	mpan 689	. . . . . 6 ⊢ ([⟨𝐶, 𝐷⟩] ∼ ∈ ((𝐺 × 𝐺) / ∼ ) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
12	11, 6	eleq2s 2862	. . . . 5 ⊢ ([⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻 → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
13		opelxp 5736	. . . . 5 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺) ↔ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))
14	12, 13	sylib 218	. . . 4 ⊢ ([⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻 → (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))
15	9, 14	anim12i 612	. . 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		brecop2.4	. . . . 5 ⊢ ≤ ⊆ (𝐺 × 𝐺)
18	17	brel 5765	. . . 4 ⊢ ((𝐴 + 𝐷) ≤ (𝐵 + 𝐶) → ((𝐴 + 𝐷) ∈ 𝐺 ∧ (𝐵 + 𝐶) ∈ 𝐺))
19		brecop2.6	. . . . . 6 ⊢ dom + = (𝐺 × 𝐺)
20		brecop2.5	. . . . . 6 ⊢ ¬ ∅ ∈ 𝐺
21	19, 20	ndmovrcl 7636	. . . . 5 ⊢ ((𝐴 + 𝐷) ∈ 𝐺 → (𝐴 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))
22	19, 20	ndmovrcl 7636	. . . . 5 ⊢ ((𝐵 + 𝐶) ∈ 𝐺 → (𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐺))
23	21, 22	anim12i 612	. . . 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 656	. . 3 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) ∧ (𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐺)) ↔ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)))
26	24, 25	sylib 218	. 2 ⊢ ((𝐴 + 𝐷) ≤ (𝐵 + 𝐶) → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)))
27		brecop2.7	. 2 ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([⟨𝐴, 𝐵⟩] ∼ 𝑅[⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) ≤ (𝐵 + 𝐶)))
28	16, 26, 27	pm5.21nii 378	1 ⊢ ([⟨𝐴, 𝐵⟩] ∼ 𝑅[⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) ≤ (𝐵 + 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  ∅c0 4352  ⟨cop 4654   class class class wbr 5166   × cxp 5698  dom cdm 5700  (class class class)co 7448  [cec 8761   / cqs 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fv 6581  df-ov 7451  df-ec 8765  df-qs 8769

This theorem is referenced by:  ltsrpr  11146