Theorem prsrlem1 11032

Description: Decomposing signed reals into positive reals. Lemma for addsrpr 11035 and mulsrpr 11036. (Contributed by Jim Kingdon, 30-Dec-2019.)

Assertion

Ref	Expression
prsrlem1	⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔))))

Distinct variable group:   𝑓,𝑔,ℎ,𝑠,𝑡,𝑢,𝑣,𝑤

Allowed substitution hints:   𝐴(𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,ℎ,𝑠)   𝐵(𝑤,𝑣,𝑢,𝑡,𝑓,𝑔,ℎ,𝑠)

Proof of Theorem prsrlem1

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrer 11023	. . . . . 6 ⊢ ~R Er (P × P)
2		erdm 8684	. . . . . 6 ⊢ ( ~R Er (P × P) → dom ~R = (P × P))
3	1, 2	ax-mp 5	. . . . 5 ⊢ dom ~R = (P × P)
4		simprll 778	. . . . . 6 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → 𝐴 = [⟨𝑤, 𝑣⟩] ~R )
5		simpll 766	. . . . . 6 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → 𝐴 ∈ ((P × P) / ~R ))
6	4, 5	eqeltrrd 2830	. . . . 5 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨𝑤, 𝑣⟩] ~R ∈ ((P × P) / ~R ))
7		ecelqsdm 8761	. . . . 5 ⊢ ((dom ~R = (P × P) ∧ [⟨𝑤, 𝑣⟩] ~R ∈ ((P × P) / ~R )) → ⟨𝑤, 𝑣⟩ ∈ (P × P))
8	3, 6, 7	sylancr 587	. . . 4 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ⟨𝑤, 𝑣⟩ ∈ (P × P))
9		opelxp 5677	. . . 4 ⊢ (⟨𝑤, 𝑣⟩ ∈ (P × P) ↔ (𝑤 ∈ P ∧ 𝑣 ∈ P))
10	8, 9	sylib 218	. . 3 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → (𝑤 ∈ P ∧ 𝑣 ∈ P))
11		simprrl 780	. . . . . 6 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → 𝐴 = [⟨𝑠, 𝑓⟩] ~R )
12	11, 5	eqeltrrd 2830	. . . . 5 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨𝑠, 𝑓⟩] ~R ∈ ((P × P) / ~R ))
13		ecelqsdm 8761	. . . . 5 ⊢ ((dom ~R = (P × P) ∧ [⟨𝑠, 𝑓⟩] ~R ∈ ((P × P) / ~R )) → ⟨𝑠, 𝑓⟩ ∈ (P × P))
14	3, 12, 13	sylancr 587	. . . 4 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ⟨𝑠, 𝑓⟩ ∈ (P × P))
15		opelxp 5677	. . . 4 ⊢ (⟨𝑠, 𝑓⟩ ∈ (P × P) ↔ (𝑠 ∈ P ∧ 𝑓 ∈ P))
16	14, 15	sylib 218	. . 3 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → (𝑠 ∈ P ∧ 𝑓 ∈ P))
17	10, 16	jca 511	. 2 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)))
18		simprlr 779	. . . . . 6 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → 𝐵 = [⟨𝑢, 𝑡⟩] ~R )
19		simplr 768	. . . . . 6 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → 𝐵 ∈ ((P × P) / ~R ))
20	18, 19	eqeltrrd 2830	. . . . 5 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨𝑢, 𝑡⟩] ~R ∈ ((P × P) / ~R ))
21		ecelqsdm 8761	. . . . 5 ⊢ ((dom ~R = (P × P) ∧ [⟨𝑢, 𝑡⟩] ~R ∈ ((P × P) / ~R )) → ⟨𝑢, 𝑡⟩ ∈ (P × P))
22	3, 20, 21	sylancr 587	. . . 4 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ⟨𝑢, 𝑡⟩ ∈ (P × P))
23		opelxp 5677	. . . 4 ⊢ (⟨𝑢, 𝑡⟩ ∈ (P × P) ↔ (𝑢 ∈ P ∧ 𝑡 ∈ P))
24	22, 23	sylib 218	. . 3 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → (𝑢 ∈ P ∧ 𝑡 ∈ P))
25		simprrr 781	. . . . . 6 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → 𝐵 = [⟨𝑔, ℎ⟩] ~R )
26	25, 19	eqeltrrd 2830	. . . . 5 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨𝑔, ℎ⟩] ~R ∈ ((P × P) / ~R ))
27		ecelqsdm 8761	. . . . 5 ⊢ ((dom ~R = (P × P) ∧ [⟨𝑔, ℎ⟩] ~R ∈ ((P × P) / ~R )) → ⟨𝑔, ℎ⟩ ∈ (P × P))
28	3, 26, 27	sylancr 587	. . . 4 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ⟨𝑔, ℎ⟩ ∈ (P × P))
29		opelxp 5677	. . . 4 ⊢ (⟨𝑔, ℎ⟩ ∈ (P × P) ↔ (𝑔 ∈ P ∧ ℎ ∈ P))
30	28, 29	sylib 218	. . 3 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → (𝑔 ∈ P ∧ ℎ ∈ P))
31	24, 30	jca 511	. 2 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P)))
32	4, 11	eqtr3d 2767	. . . . 5 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨𝑤, 𝑣⟩] ~R = [⟨𝑠, 𝑓⟩] ~R )
33	1	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ~R Er (P × P))
34	33, 8	erth 8728	. . . . 5 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → (⟨𝑤, 𝑣⟩ ~R ⟨𝑠, 𝑓⟩ ↔ [⟨𝑤, 𝑣⟩] ~R = [⟨𝑠, 𝑓⟩] ~R ))
35	32, 34	mpbird 257	. . . 4 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ⟨𝑤, 𝑣⟩ ~R ⟨𝑠, 𝑓⟩)
36		df-enr 11015	. . . . . 6 ⊢ ~R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑎∃𝑏∃𝑐∃𝑑((𝑥 = ⟨𝑎, 𝑏⟩ ∧ 𝑦 = ⟨𝑐, 𝑑⟩) ∧ (𝑎 +P 𝑑) = (𝑏 +P 𝑐)))}
37	36	ecopoveq 8794	. . . . 5 ⊢ (((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) → (⟨𝑤, 𝑣⟩ ~R ⟨𝑠, 𝑓⟩ ↔ (𝑤 +P 𝑓) = (𝑣 +P 𝑠)))
38	10, 16, 37	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → (⟨𝑤, 𝑣⟩ ~R ⟨𝑠, 𝑓⟩ ↔ (𝑤 +P 𝑓) = (𝑣 +P 𝑠)))
39	35, 38	mpbid 232	. . 3 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → (𝑤 +P 𝑓) = (𝑣 +P 𝑠))
40	18, 25	eqtr3d 2767	. . . . 5 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → [⟨𝑢, 𝑡⟩] ~R = [⟨𝑔, ℎ⟩] ~R )
41	33, 22	erth 8728	. . . . 5 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → (⟨𝑢, 𝑡⟩ ~R ⟨𝑔, ℎ⟩ ↔ [⟨𝑢, 𝑡⟩] ~R = [⟨𝑔, ℎ⟩] ~R ))
42	40, 41	mpbird 257	. . . 4 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ⟨𝑢, 𝑡⟩ ~R ⟨𝑔, ℎ⟩)
43	36	ecopoveq 8794	. . . . 5 ⊢ (((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P)) → (⟨𝑢, 𝑡⟩ ~R ⟨𝑔, ℎ⟩ ↔ (𝑢 +P ℎ) = (𝑡 +P 𝑔)))
44	24, 30, 43	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → (⟨𝑢, 𝑡⟩ ~R ⟨𝑔, ℎ⟩ ↔ (𝑢 +P ℎ) = (𝑡 +P 𝑔)))
45	42, 44	mpbid 232	. . 3 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → (𝑢 +P ℎ) = (𝑡 +P 𝑔))
46	39, 45	jca 511	. 2 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔)))
47	17, 31, 46	jca31 514	1 ⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R ∧ 𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R ∧ 𝐵 = [⟨𝑔, ℎ⟩] ~R ))) → ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4598   class class class wbr 5110   × cxp 5639  dom cdm 5641  (class class class)co 7390   Er wer 8671  [cec 8672   / cqs 8673  Pcnp 10819   +_P cpp 10821   ~_R cer 10824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442  df-er 8674  df-ec 8676  df-qs 8680  df-ni 10832  df-pli 10833  df-mi 10834  df-lti 10835  df-plpq 10868  df-mpq 10869  df-ltpq 10870  df-enq 10871  df-nq 10872  df-erq 10873  df-plq 10874  df-mq 10875  df-1nq 10876  df-rq 10877  df-ltnq 10878  df-np 10941  df-plp 10943  df-ltp 10945  df-enr 11015

This theorem is referenced by:  addsrmo  11033  mulsrmo  11034