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Theorem brecop2 7826
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 (((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 brecop2.7 . . . 4 𝑅 ⊆ (𝐻 × 𝐻)
21brel 5158 . . 3 ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] → ([⟨𝐴, 𝐵⟩] 𝐻 ∧ [⟨𝐶, 𝐷⟩] 𝐻))
3 brecop2.5 . . . . . . 7 dom = (𝐺 × 𝐺)
4 ecelqsdm 7802 . . . . . . 7 ((dom = (𝐺 × 𝐺) ∧ [⟨𝐴, 𝐵⟩] ∈ ((𝐺 × 𝐺) / )) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
53, 4mpan 705 . . . . . 6 ([⟨𝐴, 𝐵⟩] ∈ ((𝐺 × 𝐺) / ) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
6 brecop2.6 . . . . . 6 𝐻 = ((𝐺 × 𝐺) / )
75, 6eleq2s 2717 . . . . 5 ([⟨𝐴, 𝐵⟩] 𝐻 → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
8 opelxp 5136 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺) ↔ (𝐴𝐺𝐵𝐺))
97, 8sylib 208 . . . 4 ([⟨𝐴, 𝐵⟩] 𝐻 → (𝐴𝐺𝐵𝐺))
10 ecelqsdm 7802 . . . . . . 7 ((dom = (𝐺 × 𝐺) ∧ [⟨𝐶, 𝐷⟩] ∈ ((𝐺 × 𝐺) / )) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
113, 10mpan 705 . . . . . 6 ([⟨𝐶, 𝐷⟩] ∈ ((𝐺 × 𝐺) / ) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
1211, 6eleq2s 2717 . . . . 5 ([⟨𝐶, 𝐷⟩] 𝐻 → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
13 opelxp 5136 . . . . 5 (⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺) ↔ (𝐶𝐺𝐷𝐺))
1412, 13sylib 208 . . . 4 ([⟨𝐶, 𝐷⟩] 𝐻 → (𝐶𝐺𝐷𝐺))
159, 14anim12i 589 . . 3 (([⟨𝐴, 𝐵⟩] 𝐻 ∧ [⟨𝐶, 𝐷⟩] 𝐻) → ((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)))
162, 15syl 17 . 2 ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] → ((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)))
17 brecop2.8 . . . . 5 ⊆ (𝐺 × 𝐺)
1817brel 5158 . . . 4 ((𝐴 + 𝐷) (𝐵 + 𝐶) → ((𝐴 + 𝐷) ∈ 𝐺 ∧ (𝐵 + 𝐶) ∈ 𝐺))
19 brecop2.10 . . . . . 6 dom + = (𝐺 × 𝐺)
20 brecop2.9 . . . . . 6 ¬ ∅ ∈ 𝐺
2119, 20ndmovrcl 6805 . . . . 5 ((𝐴 + 𝐷) ∈ 𝐺 → (𝐴𝐺𝐷𝐺))
2219, 20ndmovrcl 6805 . . . . 5 ((𝐵 + 𝐶) ∈ 𝐺 → (𝐵𝐺𝐶𝐺))
2321, 22anim12i 589 . . . 4 (((𝐴 + 𝐷) ∈ 𝐺 ∧ (𝐵 + 𝐶) ∈ 𝐺) → ((𝐴𝐺𝐷𝐺) ∧ (𝐵𝐺𝐶𝐺)))
2418, 23syl 17 . . 3 ((𝐴 + 𝐷) (𝐵 + 𝐶) → ((𝐴𝐺𝐷𝐺) ∧ (𝐵𝐺𝐶𝐺)))
25 an42 865 . . 3 (((𝐴𝐺𝐷𝐺) ∧ (𝐵𝐺𝐶𝐺)) ↔ ((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)))
2624, 25sylib 208 . 2 ((𝐴 + 𝐷) (𝐵 + 𝐶) → ((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)))
27 brecop2.11 . 2 (((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)) → ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) (𝐵 + 𝐶)))
2816, 26, 27pm5.21nii 368 1 ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) (𝐵 + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  wss 3567  c0 3907  cop 4174   class class class wbr 4644   × cxp 5102  dom cdm 5104  (class class class)co 6635  [cec 7725   / cqs 7726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fv 5884  df-ov 6638  df-ec 7729  df-qs 7733
This theorem is referenced by:  ltsrpr  9883
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