Theorem mulextsr1 7841

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 7787	. 2 ⊢ R = ((P × P) / ~R )
2		oveq1 5925	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ))
3	2	breq1d 4039	. . 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 4032	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
5		breq2 4033	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ [⟨𝑧, 𝑤⟩] ~R <R 𝐴))
6	4, 5	orbi12d 794	. . 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 5925	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ))
9	8	breq2d 4041	. . 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 4033	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝐴 <R 𝐵))
11		breq1 4032	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R <R 𝐴 ↔ 𝐵 <R 𝐴))
12	10, 11	orbi12d 794	. . 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 5926	. . . 4 ⊢ ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐴 ·R 𝐶))
15		oveq2 5926	. . . 4 ⊢ ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐵 ·R 𝐶))
16	14, 15	breq12d 4042	. . 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 7840	. . 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 7806	. . . . . 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 1018	. . . . 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 7806	. . . . . 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 1017	. . . . 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 4042	. . . 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 1023	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑥 ∈ P)
25		simp3l 1027	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑢 ∈ P)
26		mulclpr 7632	. . . . . . 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 1024	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑦 ∈ P)
29		simp3r 1028	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑣 ∈ P)
30		mulclpr 7632	. . . . . . 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 7597	. . . . . 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 7632	. . . . . . 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 7632	. . . . . . 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 7597	. . . . . 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 1025	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑧 ∈ P)
41		mulclpr 7632	. . . . . . 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 1026	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑢 ∈ P ∧ 𝑣 ∈ P)) → 𝑤 ∈ P)
44		mulclpr 7632	. . . . . . 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 7597	. . . . . 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 7632	. . . . . . 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 7632	. . . . . . 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 7597	. . . . . 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 7807	. . . . 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 1250	. . . 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 7807	. . . . 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 1019	. . . 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 7807	. . . . . 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 1019	. . . 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 794	. . 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 6678	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 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ⟨cop 3621   class class class wbr 4029  (class class class)co 5918  [cec 6585  Pcnp 7351   +_P cpp 7353   ·_P cmp 7354  <_P cltp 7355   ~_R cer 7356  Rcnr 7357   ·_R cmr 7362   <_R cltr 7363

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-i1p 7527  df-iplp 7528  df-imp 7529  df-iltp 7530  df-enr 7786  df-nr 7787  df-mr 7789  df-ltr 7790

This theorem is referenced by:  axpre-mulext  7948