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Theorem brecop2 8751
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 (((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)) → ([⟨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
StepHypRef Expression
1 brecop2.3 . . . 4 𝑅 ⊆ (𝐻 × 𝐻)
21brel 5698 . . 3 ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] → ([⟨𝐴, 𝐵⟩] 𝐻 ∧ [⟨𝐶, 𝐷⟩] 𝐻))
3 brecop2.1 . . . . . . 7 dom = (𝐺 × 𝐺)
4 ecelqsdm 8727 . . . . . . 7 ((dom = (𝐺 × 𝐺) ∧ [⟨𝐴, 𝐵⟩] ∈ ((𝐺 × 𝐺) / )) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
53, 4mpan 689 . . . . . 6 ([⟨𝐴, 𝐵⟩] ∈ ((𝐺 × 𝐺) / ) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
6 brecop2.2 . . . . . 6 𝐻 = ((𝐺 × 𝐺) / )
75, 6eleq2s 2856 . . . . 5 ([⟨𝐴, 𝐵⟩] 𝐻 → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
8 opelxp 5670 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺) ↔ (𝐴𝐺𝐵𝐺))
97, 8sylib 217 . . . 4 ([⟨𝐴, 𝐵⟩] 𝐻 → (𝐴𝐺𝐵𝐺))
10 ecelqsdm 8727 . . . . . . 7 ((dom = (𝐺 × 𝐺) ∧ [⟨𝐶, 𝐷⟩] ∈ ((𝐺 × 𝐺) / )) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
113, 10mpan 689 . . . . . 6 ([⟨𝐶, 𝐷⟩] ∈ ((𝐺 × 𝐺) / ) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
1211, 6eleq2s 2856 . . . . 5 ([⟨𝐶, 𝐷⟩] 𝐻 → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
13 opelxp 5670 . . . . 5 (⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺) ↔ (𝐶𝐺𝐷𝐺))
1412, 13sylib 217 . . . 4 ([⟨𝐶, 𝐷⟩] 𝐻 → (𝐶𝐺𝐷𝐺))
159, 14anim12i 614 . . 3 (([⟨𝐴, 𝐵⟩] 𝐻 ∧ [⟨𝐶, 𝐷⟩] 𝐻) → ((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)))
162, 15syl 17 . 2 ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] → ((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)))
17 brecop2.4 . . . . 5 ⊆ (𝐺 × 𝐺)
1817brel 5698 . . . 4 ((𝐴 + 𝐷) (𝐵 + 𝐶) → ((𝐴 + 𝐷) ∈ 𝐺 ∧ (𝐵 + 𝐶) ∈ 𝐺))
19 brecop2.6 . . . . . 6 dom + = (𝐺 × 𝐺)
20 brecop2.5 . . . . . 6 ¬ ∅ ∈ 𝐺
2119, 20ndmovrcl 7541 . . . . 5 ((𝐴 + 𝐷) ∈ 𝐺 → (𝐴𝐺𝐷𝐺))
2219, 20ndmovrcl 7541 . . . . 5 ((𝐵 + 𝐶) ∈ 𝐺 → (𝐵𝐺𝐶𝐺))
2321, 22anim12i 614 . . . 4 (((𝐴 + 𝐷) ∈ 𝐺 ∧ (𝐵 + 𝐶) ∈ 𝐺) → ((𝐴𝐺𝐷𝐺) ∧ (𝐵𝐺𝐶𝐺)))
2418, 23syl 17 . . 3 ((𝐴 + 𝐷) (𝐵 + 𝐶) → ((𝐴𝐺𝐷𝐺) ∧ (𝐵𝐺𝐶𝐺)))
25 an42 656 . . 3 (((𝐴𝐺𝐷𝐺) ∧ (𝐵𝐺𝐶𝐺)) ↔ ((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)))
2624, 25sylib 217 . 2 ((𝐴 + 𝐷) (𝐵 + 𝐶) → ((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)))
27 brecop2.7 . 2 (((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)) → ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) (𝐵 + 𝐶)))
2816, 26, 27pm5.21nii 380 1 ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) (𝐵 + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wss 3911  c0 4283  cop 4593   class class class wbr 5106   × cxp 5632  dom cdm 5634  (class class class)co 7358  [cec 8647   / cqs 8648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fv 6505  df-ov 7361  df-ec 8651  df-qs 8655
This theorem is referenced by:  ltsrpr  11014
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