Theorem brecop2 3243

Description: Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis.

Hypotheses

Ref	Expression
brecop2.1	⊢ S ∈ V
brecop2.2	⊢ B ∈ V
brecop2.3	⊢ C ∈ V
brecop2.4	⊢ D ∈ V
brecop2.5	⊢ Er S
brecop2.6	⊢ dom S = (G × G)
brecop2.7	⊢ H = ((G × G) / S)
brecop2.8	⊢ R ⊆ (H × H)
brecop2.9	⊢ Q ⊆ (G × G)
brecop2.10	⊢ ¬ ∅ ∈ G
brecop2.11	⊢ dom F = (G × G)
brecop2.12	⊢ (((A ∈ G ∧ B ∈ G) ∧ (C ∈ G ∧ D ∈ G)) → ([⟨A, B⟩]SR[⟨C, D⟩]S ↔ (AFD)Q(BFC)))

Assertion

Ref	Expression
brecop2	⊢ ([⟨A, B⟩]SR[⟨C, D⟩]S ↔ (AFD)Q(BFC))

Proof of Theorem brecop2

Step	Hyp	Ref	Expression
1		brecop2.1	. . . . 5 ⊢ S ∈ V
2		ecexg 3204	. . . . 5 ⊢ (S ∈ V → [⟨C, D⟩]S ∈ V)
3	1, 2	ax-mp 6	. . . 4 ⊢ [⟨C, D⟩]S ∈ V
4		brecop2.8	. . . 4 ⊢ R ⊆ (H × H)
5	3, 4	brel 2459	. . 3 ⊢ ([⟨A, B⟩]SR[⟨C, D⟩]S → ([⟨A, B⟩]S ∈ H ∧ [⟨C, D⟩]S ∈ H))
6		brecop2.7	. . . . . . 7 ⊢ H = ((G × G) / S)
7	6	eleq2i 1153	. . . . . 6 ⊢ ([⟨A, B⟩]S ∈ H ↔ [⟨A, B⟩]S ∈ ((G × G) / S))
8		opex 1893	. . . . . . 7 ⊢ ⟨A, B⟩ ∈ V
9		brecop2.5	. . . . . . 7 ⊢ Er S
10		brecop2.6	. . . . . . 7 ⊢ dom S = (G × G)
11	8, 9, 10	ecelqsdm 3235	. . . . . 6 ⊢ ([⟨A, B⟩]S ∈ ((G × G) / S) → ⟨A, B⟩ ∈ (G × G))
12	7, 11	sylbi 174	. . . . 5 ⊢ ([⟨A, B⟩]S ∈ H → ⟨A, B⟩ ∈ (G × G))
13		brecop2.2	. . . . . 6 ⊢ B ∈ V
14	13	opelxp 2452	. . . . 5 ⊢ (⟨A, B⟩ ∈ (G × G) ↔ (A ∈ G ∧ B ∈ G))
15	12, 14	sylib 173	. . . 4 ⊢ ([⟨A, B⟩]S ∈ H → (A ∈ G ∧ B ∈ G))
16	6	eleq2i 1153	. . . . . 6 ⊢ ([⟨C, D⟩]S ∈ H ↔ [⟨C, D⟩]S ∈ ((G × G) / S))
17		opex 1893	. . . . . . 7 ⊢ ⟨C, D⟩ ∈ V
18	17, 9, 10	ecelqsdm 3235	. . . . . 6 ⊢ ([⟨C, D⟩]S ∈ ((G × G) / S) → ⟨C, D⟩ ∈ (G × G))
19	16, 18	sylbi 174	. . . . 5 ⊢ ([⟨C, D⟩]S ∈ H → ⟨C, D⟩ ∈ (G × G))
20		brecop2.4	. . . . . 6 ⊢ D ∈ V
21	20	opelxp 2452	. . . . 5 ⊢ (⟨C, D⟩ ∈ (G × G) ↔ (C ∈ G ∧ D ∈ G))
22	19, 21	sylib 173	. . . 4 ⊢ ([⟨C, D⟩]S ∈ H → (C ∈ G ∧ D ∈ G))
23	15, 22	anim12i 268	. . 3 ⊢ (([⟨A, B⟩]S ∈ H ∧ [⟨C, D⟩]S ∈ H) → ((A ∈ G ∧ B ∈ G) ∧ (C ∈ G ∧ D ∈ G)))
24	5, 23	syl 12	. 2 ⊢ ([⟨A, B⟩]SR[⟨C, D⟩]S → ((A ∈ G ∧ B ∈ G) ∧ (C ∈ G ∧ D ∈ G)))
25		oprex 3018	. . . . 5 ⊢ (BFC) ∈ V
26		brecop2.9	. . . . 5 ⊢ Q ⊆ (G × G)
27	25, 26	brel 2459	. . . 4 ⊢ ((AFD)Q(BFC) → ((AFD) ∈ G ∧ (BFC) ∈ G))
28		brecop2.11	. . . . . 6 ⊢ dom F = (G × G)
29		brecop2.10	. . . . . 6 ⊢ ¬ ∅ ∈ G
30	20, 28, 29	ndmoprrcl 3060	. . . . 5 ⊢ ((AFD) ∈ G → (A ∈ G ∧ D ∈ G))
31		brecop2.3	. . . . . 6 ⊢ C ∈ V
32	31, 28, 29	ndmoprrcl 3060	. . . . 5 ⊢ ((BFC) ∈ G → (B ∈ G ∧ C ∈ G))
33	30, 32	anim12i 268	. . . 4 ⊢ (((AFD) ∈ G ∧ (BFC) ∈ G) → ((A ∈ G ∧ D ∈ G) ∧ (B ∈ G ∧ C ∈ G)))
34	27, 33	syl 12	. . 3 ⊢ ((AFD)Q(BFC) → ((A ∈ G ∧ D ∈ G) ∧ (B ∈ G ∧ C ∈ G)))
35		an42 389	. . 3 ⊢ (((A ∈ G ∧ D ∈ G) ∧ (B ∈ G ∧ C ∈ G)) ↔ ((A ∈ G ∧ B ∈ G) ∧ (C ∈ G ∧ D ∈ G)))
36	34, 35	sylib 173	. 2 ⊢ ((AFD)Q(BFC) → ((A ∈ G ∧ B ∈ G) ∧ (C ∈ G ∧ D ∈ G)))
37		brecop2.12	. 2 ⊢ (((A ∈ G ∧ B ∈ G) ∧ (C ∈ G ∧ D ∈ G)) → ([⟨A, B⟩]SR[⟨C, D⟩]S ↔ (AFD)Q(BFC)))
38	24, 36, 37	pm5.21nii 504	1 ⊢ ([⟨A, B⟩]SR[⟨C, D⟩]S ↔ (AFD)Q(BFC))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ⟨cop 1810   class class class wbr 2054   × cxp 2408  dom cdm 2410  (class class class)co 3001  Er wer 3197  [cec 3198   / cqs 3199

This theorem is referenced by:  ordpipq 3850  ltsrpr 3980

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-ec 3202  df-qs 3205

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