Theorem mulextsr1 8092

Description: Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)

Assertion

Ref	Expression
mulextsr1	⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶) → (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)))

Proof of Theorem mulextsr1

Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 8038	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 6056	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ))
3	2	breq1d 4118	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R )))
4		breq1 4111	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
5		breq2 4112	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ [⟨𝑧, 𝑤⟩] ~R <R 𝐴))
6	4, 5	orbi12d 801	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴)))
7	3, 6	imbi12d 234	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ((([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R )) ↔ ((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴))))
8		oveq1 6056	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ))
9	8	breq2d 4120	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R )))
10		breq2 4112	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R 𝐵))
11		breq1 4111	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R <R 𝐴 ↔ 𝐵 <R 𝐴))
12	10, 11	orbi12d 801	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) ↔ (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)))
13	9, 12	imbi12d 234	. 2 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴)) ↔ ((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) → (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴))))
14		oveq2 6057	. . . 4 ⊢ ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐴 ·R 𝐶))
15		oveq2 6057	. . . 4 ⊢ ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐵 ·R 𝐶))
16	14, 15	breq12d 4121	. . 3 ⊢ ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → ((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ (𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶)))
17	16	imbi1d 231	. 2 ⊢ ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) → (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)) ↔ ((𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶) → (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴))))
18		mulextsr1lem 8091	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ((((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) +P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)))<P (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) +P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣))) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))))
19		mulsrpr 8057	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = [⟨((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)), ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))⟩] ~R )
20	19	3adant2 1043	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = [⟨((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)), ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))⟩] ~R )
21		mulsrpr 8057	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = [⟨((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)), ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))⟩] ~R )
22	21	3adant1 1042	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = [⟨((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)), ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))⟩] ~R )
23	20, 22	breq12d 4121	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ [⟨((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)), ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))⟩] ~R <R [⟨((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)), ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))⟩] ~R ))
24		simp1l 1048	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑥 ∈ P)
25		simp3l 1052	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑢 ∈ P)
26		mulclpr 7883	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑢 ∈ P) → (𝑥 ·P 𝑢) ∈ P)
27	24, 25, 26	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (𝑥 ·P 𝑢) ∈ P)
28		simp1r 1049	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑦 ∈ P)
29		simp3r 1053	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑣 ∈ P)
30		mulclpr 7883	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑣 ∈ P) → (𝑦 ·P 𝑣) ∈ P)
31	28, 29, 30	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (𝑦 ·P 𝑣) ∈ P)
32		addclpr 7848	. . . . . 6 ⊢ (((𝑥 ·P 𝑢) ∈ P ∧ (𝑦 ·P 𝑣) ∈ P) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
33	27, 31, 32	syl2anc 411	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
34		mulclpr 7883	. . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 𝑣 ∈ P) → (𝑥 ·P 𝑣) ∈ P)
35	24, 29, 34	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (𝑥 ·P 𝑣) ∈ P)
36		mulclpr 7883	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑢 ∈ P) → (𝑦 ·P 𝑢) ∈ P)
37	28, 25, 36	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (𝑦 ·P 𝑢) ∈ P)
38		addclpr 7848	. . . . . 6 ⊢ (((𝑥 ·P 𝑣) ∈ P ∧ (𝑦 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
39	35, 37, 38	syl2anc 411	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
40		simp2l 1050	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑧 ∈ P)
41		mulclpr 7883	. . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 𝑢 ∈ P) → (𝑧 ·P 𝑢) ∈ P)
42	40, 25, 41	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (𝑧 ·P 𝑢) ∈ P)
43		simp2r 1051	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑤 ∈ P)
44		mulclpr 7883	. . . . . . 7 ⊢ ((𝑤 ∈ P ∧ 𝑣 ∈ P) → (𝑤 ·P 𝑣) ∈ P)
45	43, 29, 44	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (𝑤 ·P 𝑣) ∈ P)
46		addclpr 7848	. . . . . 6 ⊢ (((𝑧 ·P 𝑢) ∈ P ∧ (𝑤 ·P 𝑣) ∈ P) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
47	42, 45, 46	syl2anc 411	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
48		mulclpr 7883	. . . . . . 7 ⊢ ((𝑧 ∈ P ∧ 𝑣 ∈ P) → (𝑧 ·P 𝑣) ∈ P)
49	40, 29, 48	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (𝑧 ·P 𝑣) ∈ P)
50		mulclpr 7883	. . . . . . 7 ⊢ ((𝑤 ∈ P ∧ 𝑢 ∈ P) → (𝑤 ·P 𝑢) ∈ P)
51	43, 25, 50	syl2anc 411	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (𝑤 ·P 𝑢) ∈ P)
52		addclpr 7848	. . . . . 6 ⊢ (((𝑧 ·P 𝑣) ∈ P ∧ (𝑤 ·P 𝑢) ∈ P) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
53	49, 51, 52	syl2anc 411	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
54		ltsrprg 8058	. . . . 5 ⊢ (((((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P ∧ ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P) ∧ (((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P ∧ ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)) → ([⟨((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)), ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))⟩] ~R <R [⟨((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)), ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))⟩] ~R ↔ (((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) +P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)))<P (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) +P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)))))
55	33, 39, 47, 53, 54	syl22anc 1275	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ([⟨((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)), ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))⟩] ~R <R [⟨((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)), ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))⟩] ~R ↔ (((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) +P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)))<P (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) +P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)))))
56	23, 55	bitrd 188	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ (((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) +P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)))<P (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) +P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)))))
57		ltsrprg 8058	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
58	57	3adant3 1044	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
59		ltsrprg 8058	. . . . . 6 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
60	59	ancoms 268	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
61	60	3adant3 1044	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
62	58, 61	orbi12d 801	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))))
63	18, 56, 62	3imtr4d 203	. 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → (([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R )))
64	1, 7, 13, 17, 63	3ecoptocl 6857	1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R ∧ 𝐶 ∈ R) → ((𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶) → (𝐴 <R 𝐵 ∨ 𝐵 <R 𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ⟨cop 3691   class class class wbr 4108  (class class class)co 6049  [cec 6764  Pcnp 7602   +_P cpp 7604   ·_P cmp 7605  <_P cltp 7606   ~_R cer 7607  Rcnr 7608   ·_R cmr 7613   <_R cltr 7614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7615  df-pli 7616  df-mi 7617  df-lti 7618  df-plpq 7655  df-mpq 7656  df-enq 7658  df-nqqs 7659  df-plqqs 7660  df-mqqs 7661  df-1nqqs 7662  df-rq 7663  df-ltnqqs 7664  df-enq0 7735  df-nq0 7736  df-0nq0 7737  df-plq0 7738  df-mq0 7739  df-inp 7777  df-i1p 7778  df-iplp 7779  df-imp 7780  df-iltp 7781  df-enr 8037  df-nr 8038  df-mr 8040  df-ltr 8041

This theorem is referenced by:  axpre-mulext  8199