Theorem mulgt0sr 6862

Description: The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)

Assertion

Ref	Expression
mulgt0sr	⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))

Proof of Theorem mulgt0sr

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

Step	Hyp	Ref	Expression
1		ltrelsr 6823	. . . . 5 ⊢ <R ⊆ (R × R)
2	1	brel 4392	. . . 4 ⊢ (0R <R 𝐴 → (0R ∈ R ∧ 𝐴 ∈ R))
3	2	simprd 107	. . 3 ⊢ (0R <R 𝐴 → 𝐴 ∈ R)
4	1	brel 4392	. . . 4 ⊢ (0R <R 𝐵 → (0R ∈ R ∧ 𝐵 ∈ R))
5	4	simprd 107	. . 3 ⊢ (0R <R 𝐵 → 𝐵 ∈ R)
6	3, 5	anim12i 321	. 2 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → (𝐴 ∈ R ∧ 𝐵 ∈ R))
7		df-nr 6812	. . 3 ⊢ R = ((P × P) / ~R )
8		breq2 3768	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 0R <R 𝐴))
9	8	anbi1d 438	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R )))
10		oveq1 5519	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))
11	10	breq2d 3776	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R )))
12	9, 11	imbi12d 223	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )) ↔ ((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) → 0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))))
13		breq2 3768	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R [⟨𝑧, 𝑤⟩] ~R ↔ 0R <R 𝐵))
14	13	anbi2d 437	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (0R <R 𝐴 ∧ 0R <R 𝐵)))
15		oveq2 5520	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R 𝐵))
16	15	breq2d 3776	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R 𝐵)))
17	14, 16	imbi12d 223	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) → 0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R )) ↔ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))))
18		gt0srpr 6833	. . . . 5 ⊢ (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 𝑦<P 𝑥)
19		gt0srpr 6833	. . . . 5 ⊢ (0R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑤<P 𝑧)
20	18, 19	anbi12i 433	. . . 4 ⊢ ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝑦<P 𝑥 ∧ 𝑤<P 𝑧))
21		ltexpri 6711	. . . . . . 7 ⊢ (𝑦<P 𝑥 → ∃𝑣 ∈ P (𝑦 +P 𝑣) = 𝑥)
22		ltexpri 6711	. . . . . . . . 9 ⊢ (𝑤<P 𝑧 → ∃𝑢 ∈ P (𝑤 +P 𝑢) = 𝑧)
23		addclpr 6635	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) ∈ P)
24	23	adantl 262	. . . . . . . . . . . . 13 ⊢ ((((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) ∈ P)
25		simplrr 488	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 +P 𝑣) = 𝑥)
26		simplr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → 𝑦 ∈ P)
27	26	ad2antrr 457	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑦 ∈ P)
28		simplrl 487	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑣 ∈ P)
29	24, 27, 28	caovcld 5654	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 +P 𝑣) ∈ P)
30	25, 29	eqeltrrd 2115	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑥 ∈ P)
31		simplrr 488	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) → 𝑤 ∈ P)
32	31	adantr 261	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑤 ∈ P)
33		mulclpr 6670	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 ·P 𝑤) ∈ P)
34	30, 32, 33	syl2anc 391	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑥 ·P 𝑤) ∈ P)
35		simplrl 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) → 𝑧 ∈ P)
36	35	adantr 261	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑧 ∈ P)
37		mulclpr 6670	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 ·P 𝑧) ∈ P)
38	27, 36, 37	syl2anc 391	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P 𝑧) ∈ P)
39	24, 34, 38	caovcld 5654	. . . . . . . . . . . 12 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)
40		simprl 483	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → 𝑢 ∈ P)
41		mulclpr 6670	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ P ∧ 𝑢 ∈ P) → (𝑣 ·P 𝑢) ∈ P)
42	28, 40, 41	syl2anc 391	. . . . . . . . . . . 12 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑣 ·P 𝑢) ∈ P)
43		ltaddpr 6695	. . . . . . . . . . . 12 ⊢ ((((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P ∧ (𝑣 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)))
44	39, 42, 43	syl2anc 391	. . . . . . . . . . 11 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)))
45		simprr 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑤 +P 𝑢) = 𝑧)
46		oveq12 5521	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (𝑥 ·P 𝑧))
47	46	oveq1d 5527	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → (((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))) = ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))))
48	25, 45, 47	syl2anc 391	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))) = ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))))
49		distrprg 6686	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P ∧ 𝑢 ∈ P) → (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)))
50	27, 32, 40, 49	syl3anc 1135	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P (𝑤 +P 𝑢)) = ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)))
51		oveq2 5520	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 +P 𝑢) = 𝑧 → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
52	51	adantl 262	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧) → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
53	52	adantl 262	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
54	50, 53	eqtr3d 2074	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) = (𝑦 ·P 𝑧))
55	54	oveq1d 5527	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢))) = ((𝑦 ·P 𝑧) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢))))
56		distrprg 6686	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓 ·P (𝑔 +P ℎ)) = ((𝑓 ·P 𝑔) +P (𝑓 ·P ℎ)))
57	56	adantl 262	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → (𝑓 ·P (𝑔 +P ℎ)) = ((𝑓 ·P 𝑔) +P (𝑓 ·P ℎ)))
58		mulcomprg 6678	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓))
59	58	adantl 262	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓))
60	57, 27, 28, 32, 24, 59	caovdir2d 5677	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P 𝑤) = ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)))
61	57, 27, 28, 40, 24, 59	caovdir2d 5677	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P 𝑢) = ((𝑦 ·P 𝑢) +P (𝑣 ·P 𝑢)))
62	60, 61	oveq12d 5530	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑦 +P 𝑣) ·P 𝑤) +P ((𝑦 +P 𝑣) ·P 𝑢)) = (((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)) +P ((𝑦 ·P 𝑢) +P (𝑣 ·P 𝑢))))
63		distrprg 6686	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 +P 𝑣) ∈ P ∧ 𝑤 ∈ P ∧ 𝑢 ∈ P) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 +P 𝑣) ·P 𝑤) +P ((𝑦 +P 𝑣) ·P 𝑢)))
64	29, 32, 40, 63	syl3anc 1135	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 +P 𝑣) ·P 𝑤) +P ((𝑦 +P 𝑣) ·P 𝑢)))
65		mulclpr 6670	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P)
66	27, 32, 65	syl2anc 391	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P 𝑤) ∈ P)
67		mulclpr 6670	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ P ∧ 𝑢 ∈ P) → (𝑦 ·P 𝑢) ∈ P)
68	27, 40, 67	syl2anc 391	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑦 ·P 𝑢) ∈ P)
69		mulclpr 6670	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ P ∧ 𝑤 ∈ P) → (𝑣 ·P 𝑤) ∈ P)
70	28, 32, 69	syl2anc 391	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑣 ·P 𝑤) ∈ P)
71		addcomprg 6676	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
72	71	adantl 262	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
73		addassprg 6677	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ)))
74	73	adantl 262	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ)))
75	66, 68, 70, 72, 74, 42, 24	caov4d 5685	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢))) = (((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)) +P ((𝑦 ·P 𝑢) +P (𝑣 ·P 𝑢))))
76	62, 64, 75	3eqtr4d 2082	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (((𝑦 ·P 𝑤) +P (𝑦 ·P 𝑢)) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢))))
77	70, 38, 42, 72, 74	caov12d 5682	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) = ((𝑦 ·P 𝑧) +P ((𝑣 ·P 𝑤) +P (𝑣 ·P 𝑢))))
78	55, 76, 77	3eqtr4d 2082	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = ((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))))
79		oveq1 5519	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
80	79	adantl 262	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥) → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
81	80	ad2antlr 458	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
82	60, 81	eqtr3d 2074	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤)) = (𝑥 ·P 𝑤))
83	78, 82	oveq12d 5530	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))) = (((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) +P (𝑥 ·P 𝑤)))
84	48, 83	eqtr3d 2074	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))) = (((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) +P (𝑥 ·P 𝑤)))
85		mulclpr 6670	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P)
86	30, 36, 85	syl2anc 391	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (𝑥 ·P 𝑧) ∈ P)
87		addassprg 6677	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P ∧ (𝑣 ·P 𝑤) ∈ P) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))))
88	86, 66, 70, 87	syl3anc 1135	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))))
89		addclpr 6635	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
90	86, 66, 89	syl2anc 391	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
91		addcomprg 6676	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ (𝑣 ·P 𝑤) ∈ P) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
92	90, 70, 91	syl2anc 391	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) +P (𝑣 ·P 𝑤)) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
93	88, 92	eqtr3d 2074	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑧) +P ((𝑦 ·P 𝑤) +P (𝑣 ·P 𝑤))) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
94	24, 38, 42	caovcld 5654	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)) ∈ P)
95		addassprg 6677	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ·P 𝑤) ∈ P ∧ (𝑥 ·P 𝑤) ∈ P ∧ ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)) ∈ P) → (((𝑣 ·P 𝑤) +P (𝑥 ·P 𝑤)) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)))))
96	70, 34, 94, 95	syl3anc 1135	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑣 ·P 𝑤) +P (𝑥 ·P 𝑤)) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)))))
97	70, 94, 34, 72, 74	caov32d 5681	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) +P (𝑥 ·P 𝑤)) = (((𝑣 ·P 𝑤) +P (𝑥 ·P 𝑤)) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))))
98		addassprg 6677	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P ∧ (𝑣 ·P 𝑢) ∈ P) → (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) = ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))))
99	34, 38, 42, 98	syl3anc 1135	. . . . . . . . . . . . . . 15 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) = ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))))
100	99	oveq2d 5528	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑣 ·P 𝑤) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))) = ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢)))))
101	96, 97, 100	3eqtr4d 2082	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑣 ·P 𝑤) +P ((𝑦 ·P 𝑧) +P (𝑣 ·P 𝑢))) +P (𝑥 ·P 𝑤)) = ((𝑣 ·P 𝑤) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))))
102	84, 93, 101	3eqtr3d 2080	. . . . . . . . . . . 12 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))) = ((𝑣 ·P 𝑤) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))))
103	24, 39, 42	caovcld 5654	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) ∈ P)
104		addcanprg 6714	. . . . . . . . . . . . 13 ⊢ (((𝑣 ·P 𝑤) ∈ P ∧ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)) ∈ P) → (((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))) = ((𝑣 ·P 𝑤) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))))
105	70, 90, 103, 104	syl3anc 1135	. . . . . . . . . . . 12 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → (((𝑣 ·P 𝑤) +P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))) = ((𝑣 ·P 𝑤) +P (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢))))
106	102, 105	mpd 13	. . . . . . . . . . 11 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = (((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) +P (𝑣 ·P 𝑢)))
107	44, 106	breqtrrd 3790	. . . . . . . . . 10 ⊢ (((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) ∧ (𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))
108	107	rexlimdvaa 2434	. . . . . . . . 9 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) → (∃𝑢 ∈ P (𝑤 +P 𝑢) = 𝑧 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
109	22, 108	syl5 28	. . . . . . . 8 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) → (𝑤<P 𝑧 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
110	109	rexlimdvaa 2434	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (∃𝑣 ∈ P (𝑦 +P 𝑣) = 𝑥 → (𝑤<P 𝑧 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
111	21, 110	syl5 28	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦<P 𝑥 → (𝑤<P 𝑧 → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))))
112	111	impd 242	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑦<P 𝑥 ∧ 𝑤<P 𝑧) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
113		mulsrpr 6831	. . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = [⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R )
114	113	breq2d 3776	. . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R [⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R ))
115		gt0srpr 6833	. . . . . 6 ⊢ (0R <R [⟨((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))⟩] ~R ↔ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)))
116	114, 115	syl6bb 185	. . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))<P ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤))))
117	112, 116	sylibrd 158	. . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑦<P 𝑥 ∧ 𝑤<P 𝑧) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )))
118	20, 117	syl5bi 141	. . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) → 0R <R ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R )))
119	7, 12, 17, 118	2ecoptocl 6194	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵)))
120	6, 119	mpcom 32	1 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764  (class class class)co 5512  [cec 6104  Pcnp 6389   +_P cpp 6391   ·_P cmp 6392  <_P cltp 6393   ~_R cer 6394  Rcnr 6395  0_Rc0r 6396   ·_R cmr 6400   <_R cltr 6401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-imp 6567  df-iltp 6568  df-enr 6811  df-nr 6812  df-mr 6814  df-ltr 6815  df-0r 6816

This theorem is referenced by:  axpre-mulgt0  6961