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Theorem brecop2 8044
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 5336 . . 3 ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] → ([⟨𝐴, 𝐵⟩] 𝐻 ∧ [⟨𝐶, 𝐷⟩] 𝐻))
3 brecop2.1 . . . . . . 7 dom = (𝐺 × 𝐺)
4 ecelqsdm 8020 . . . . . . 7 ((dom = (𝐺 × 𝐺) ∧ [⟨𝐴, 𝐵⟩] ∈ ((𝐺 × 𝐺) / )) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
53, 4mpan 681 . . . . . 6 ([⟨𝐴, 𝐵⟩] ∈ ((𝐺 × 𝐺) / ) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
6 brecop2.2 . . . . . 6 𝐻 = ((𝐺 × 𝐺) / )
75, 6eleq2s 2862 . . . . 5 ([⟨𝐴, 𝐵⟩] 𝐻 → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
8 opelxp 5313 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺) ↔ (𝐴𝐺𝐵𝐺))
97, 8sylib 209 . . . 4 ([⟨𝐴, 𝐵⟩] 𝐻 → (𝐴𝐺𝐵𝐺))
10 ecelqsdm 8020 . . . . . . 7 ((dom = (𝐺 × 𝐺) ∧ [⟨𝐶, 𝐷⟩] ∈ ((𝐺 × 𝐺) / )) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
113, 10mpan 681 . . . . . 6 ([⟨𝐶, 𝐷⟩] ∈ ((𝐺 × 𝐺) / ) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
1211, 6eleq2s 2862 . . . . 5 ([⟨𝐶, 𝐷⟩] 𝐻 → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
13 opelxp 5313 . . . . 5 (⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺) ↔ (𝐶𝐺𝐷𝐺))
1412, 13sylib 209 . . . 4 ([⟨𝐶, 𝐷⟩] 𝐻 → (𝐶𝐺𝐷𝐺))
159, 14anim12i 606 . . 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 5336 . . . 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 7018 . . . . 5 ((𝐴 + 𝐷) ∈ 𝐺 → (𝐴𝐺𝐷𝐺))
2219, 20ndmovrcl 7018 . . . . 5 ((𝐵 + 𝐶) ∈ 𝐺 → (𝐵𝐺𝐶𝐺))
2321, 22anim12i 606 . . . 4 (((𝐴 + 𝐷) ∈ 𝐺 ∧ (𝐵 + 𝐶) ∈ 𝐺) → ((𝐴𝐺𝐷𝐺) ∧ (𝐵𝐺𝐶𝐺)))
2418, 23syl 17 . . 3 ((𝐴 + 𝐷) (𝐵 + 𝐶) → ((𝐴𝐺𝐷𝐺) ∧ (𝐵𝐺𝐶𝐺)))
25 an42 647 . . 3 (((𝐴𝐺𝐷𝐺) ∧ (𝐵𝐺𝐶𝐺)) ↔ ((𝐴𝐺𝐵𝐺) ∧ (𝐶𝐺𝐷𝐺)))
2624, 25sylib 209 . 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 369 1 ([⟨𝐴, 𝐵⟩] 𝑅[⟨𝐶, 𝐷⟩] ↔ (𝐴 + 𝐷) (𝐵 + 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wss 3732  c0 4079  cop 4340   class class class wbr 4809   × cxp 5275  dom cdm 5277  (class class class)co 6842  [cec 7945   / cqs 7946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-xp 5283  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fv 6076  df-ov 6845  df-ec 7949  df-qs 7953
This theorem is referenced by:  ltsrpr  10151
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