Theorem distrpqlem 3860

Description: Lemma for distributive law: cancellation of common factor.

Hypotheses

Ref	Expression
distrpqlem.1	⊢ A ∈ V
distrpqlem.2	⊢ B ∈ V
distrpqlem.3	⊢ C ∈ V

Assertion

Ref	Expression
distrpqlem	⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → [⟨(A ·N B), (A ·N C)⟩] ~Q = [⟨B, C⟩] ~Q )

Proof of Theorem distrpqlem

Step	Hyp	Ref	Expression
1		distrpqlem.1	. . . 4 ⊢ A ∈ V
2		distrpqlem.2	. . . 4 ⊢ B ∈ V
3		distrpqlem.3	. . . 4 ⊢ C ∈ V
4		visset 1350	. . . . 5 ⊢ x ∈ V
5		visset 1350	. . . . 5 ⊢ y ∈ V
6	4, 5	mulcompi 3818	. . . 4 ⊢ (x ·N y) = (y ·N x)
7		visset 1350	. . . . 5 ⊢ z ∈ V
8	5, 7	mulasspi 3819	. . . 4 ⊢ ((x ·N y) ·N z) = (x ·N (y ·N z))
9	1, 2, 3, 6, 8	caopr32 3074	. . 3 ⊢ ((A ·N B) ·N C) = ((A ·N C) ·N B)
10		mulclpi 3815	. . . . . . 7 ⊢ ((A ∈ N ∧ B ∈ N) → (A ·N B) ∈ N)
11		mulclpi 3815	. . . . . . 7 ⊢ ((A ∈ N ∧ C ∈ N) → (A ·N C) ∈ N)
12	10, 11	anim12i 268	. . . . . 6 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (A ∈ N ∧ C ∈ N)) → ((A ·N B) ∈ N ∧ (A ·N C) ∈ N))
13		pm3.27 260	. . . . . . 7 ⊢ (((A ∈ N ∧ A ∈ N) ∧ (B ∈ N ∧ C ∈ N)) → (B ∈ N ∧ C ∈ N))
14	13	an4s 390	. . . . . 6 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (A ∈ N ∧ C ∈ N)) → (B ∈ N ∧ C ∈ N))
15	12, 14	jca 236	. . . . 5 ⊢ (((A ∈ N ∧ B ∈ N) ∧ (A ∈ N ∧ C ∈ N)) → (((A ·N B) ∈ N ∧ (A ·N C) ∈ N) ∧ (B ∈ N ∧ C ∈ N)))
16	15	3impdi 630	. . . 4 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → (((A ·N B) ∈ N ∧ (A ·N C) ∈ N) ∧ (B ∈ N ∧ C ∈ N)))
17		enqbreq 3838	. . . 4 ⊢ ((((A ·N B) ∈ N ∧ (A ·N C) ∈ N) ∧ (B ∈ N ∧ C ∈ N)) → (⟨(A ·N B), (A ·N C)⟩ ~Q ⟨B, C⟩ ↔ ((A ·N B) ·N C) = ((A ·N C) ·N B)))
18	16, 17	syl 12	. . 3 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → (⟨(A ·N B), (A ·N C)⟩ ~Q ⟨B, C⟩ ↔ ((A ·N B) ·N C) = ((A ·N C) ·N B)))
19	9, 18	mpbiri 169	. 2 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → ⟨(A ·N B), (A ·N C)⟩ ~Q ⟨B, C⟩)
20		opex 1893	. . 3 ⊢ ⟨(A ·N B), (A ·N C)⟩ ∈ V
21		opex 1893	. . 3 ⊢ ⟨B, C⟩ ∈ V
22		enqer 3840	. . 3 ⊢ Er ~Q
23	20, 21, 22	erthi 3218	. 2 ⊢ (⟨(A ·N B), (A ·N C)⟩ ~Q ⟨B, C⟩ → [⟨(A ·N B), (A ·N C)⟩] ~Q = [⟨B, C⟩] ~Q )
24	19, 23	syl 12	1 ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → [⟨(A ·N B), (A ·N C)⟩] ~Q = [⟨B, C⟩] ~Q )

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  (class class class)co 3001  [cec 3198  Ncnpi 3766   ·_N cmi 3768   ~_Q ceq 3772

This theorem is referenced by:  distrpq 3861  1qec 3862  mulidpq 3863  ltexpq 3874  halfpq 3876  prlem934b 3932

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-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-ni 3794  df-mi 3796  df-enq 3831

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