Theorem mulgt0sr 7862

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 7822	. . . . 5 ⊢ <R ⊆ (R × R)
2	1	brel 4716	. . . 4 ⊢ (0R <R 𝐴 → (0R ∈ R ∧ 𝐴 ∈ R))
3	2	simprd 114	. . 3 ⊢ (0R <R 𝐴 → 𝐴 ∈ R)
4	1	brel 4716	. . . 4 ⊢ (0R <R 𝐵 → (0R ∈ R ∧ 𝐵 ∈ R))
5	4	simprd 114	. . 3 ⊢ (0R <R 𝐵 → 𝐵 ∈ R)
6	3, 5	anim12i 338	. 2 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → (𝐴 ∈ R ∧ 𝐵 ∈ R))
7		df-nr 7811	. . 3 ⊢ R = ((P × P) / ~R )
8		breq2 4038	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 0R <R 𝐴))
9	8	anbi1d 465	. . . 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 5932	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))
11	10	breq2d 4046	. . . 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 234	. . 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 4038	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R [⟨𝑧, 𝑤⟩] ~R ↔ 0R <R 𝐵))
14	13	anbi2d 464	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((0R <R 𝐴 ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (0R <R 𝐴 ∧ 0R <R 𝐵)))
15		oveq2 5933	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R 𝐵))
16	15	breq2d 4046	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (0R <R (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) ↔ 0R <R (𝐴 ·R 𝐵)))
17	14, 16	imbi12d 234	. . 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 7832	. . . . 5 ⊢ (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 𝑦<P 𝑥)
19		gt0srpr 7832	. . . . 5 ⊢ (0R <R [⟨𝑧, 𝑤⟩] ~R ↔ 𝑤<P 𝑧)
20	18, 19	anbi12i 460	. . . 4 ⊢ ((0R <R [⟨𝑥, 𝑦⟩] ~R ∧ 0R <R [⟨𝑧, 𝑤⟩] ~R ) ↔ (𝑦<P 𝑥 ∧ 𝑤<P 𝑧))
21		ltexpri 7697	. . . . . . 7 ⊢ (𝑦<P 𝑥 → ∃𝑣 ∈ P (𝑦 +P 𝑣) = 𝑥)
22		ltexpri 7697	. . . . . . . . 9 ⊢ (𝑤<P 𝑧 → ∃𝑢 ∈ P (𝑤 +P 𝑢) = 𝑧)
23		addclpr 7621	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) ∈ P)
24	23	adantl 277	. . . . . . . . . . . . 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 536	. . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → 𝑦 ∈ P)
27	26	ad2antrr 488	. . . . . . . . . . . . . . . 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 535	. . . . . . . . . . . . . . . 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 6081	. . . . . . . . . . . . . . 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 2274	. . . . . . . . . . . . . 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 536	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) → 𝑤 ∈ P)
32	31	adantr 276	. . . . . . . . . . . . . 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 7656	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 ·P 𝑤) ∈ P)
34	30, 32, 33	syl2anc 411	. . . . . . . . . . . . 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 535	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) ∧ (𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥)) → 𝑧 ∈ P)
36	35	adantr 276	. . . . . . . . . . . . . 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 7656	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 ·P 𝑧) ∈ P)
38	27, 36, 37	syl2anc 411	. . . . . . . . . . . . 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 6081	. . . . . . . . . . . 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 529	. . . . . . . . . . . . 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 7656	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ P ∧ 𝑢 ∈ P) → (𝑣 ·P 𝑢) ∈ P)
42	28, 40, 41	syl2anc 411	. . . . . . . . . . . 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 7681	. . . . . . . . . . . 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 411	. . . . . . . . . . 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 531	. . . . . . . . . . . . . . 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 5934	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (𝑥 ·P 𝑧))
47	46	oveq1d 5940	. . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . 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 7672	. . . . . . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . . . . . 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 5933	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 +P 𝑢) = 𝑧 → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
52	51	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ P ∧ (𝑤 +P 𝑢) = 𝑧) → (𝑦 ·P (𝑤 +P 𝑢)) = (𝑦 ·P 𝑧))
53	52	adantl 277	. . . . . . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . . . . . 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 5940	. . . . . . . . . . . . . . . 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 7672	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓 ·P (𝑔 +P ℎ)) = ((𝑓 ·P 𝑔) +P (𝑓 ·P ℎ)))
57	56	adantl 277	. . . . . . . . . . . . . . . . . . 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 7664	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 ·P 𝑔) = (𝑔 ·P 𝑓))
59	58	adantl 277	. . . . . . . . . . . . . . . . . . 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 6104	. . . . . . . . . . . . . . . . . 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 6104	. . . . . . . . . . . . . . . . . 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 5943	. . . . . . . . . . . . . . . . 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 7672	. . . . . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . . . . 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 7656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P)
66	27, 32, 65	syl2anc 411	. . . . . . . . . . . . . . . . . 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 7656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ P ∧ 𝑢 ∈ P) → (𝑦 ·P 𝑢) ∈ P)
68	27, 40, 67	syl2anc 411	. . . . . . . . . . . . . . . . . 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 7656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ P ∧ 𝑤 ∈ P) → (𝑣 ·P 𝑤) ∈ P)
70	28, 32, 69	syl2anc 411	. . . . . . . . . . . . . . . . . 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 7662	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
72	71	adantl 277	. . . . . . . . . . . . . . . . . 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 7663	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → ((𝑓 +P 𝑔) +P ℎ) = (𝑓 +P (𝑔 +P ℎ)))
74	73	adantl 277	. . . . . . . . . . . . . . . . . 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 6112	. . . . . . . . . . . . . . . . 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 2239	. . . . . . . . . . . . . . . 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 6109	. . . . . . . . . . . . . . . 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 2239	. . . . . . . . . . . . . . 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 5932	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 +P 𝑣) = 𝑥 → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
80	79	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ P ∧ (𝑦 +P 𝑣) = 𝑥) → ((𝑦 +P 𝑣) ·P 𝑤) = (𝑥 ·P 𝑤))
81	80	ad2antlr 489	. . . . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . . . 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 5943	. . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . 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 7656	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P)
86	30, 36, 85	syl2anc 411	. . . . . . . . . . . . . . 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 7663	. . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . 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 7621	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P)
90	86, 66, 89	syl2anc 411	. . . . . . . . . . . . . . 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 7662	. . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . 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 2231	. . . . . . . . . . . . 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 6081	. . . . . . . . . . . . . . 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 7663	. . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . 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 6108	. . . . . . . . . . . . . 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 7663	. . . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . . 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 5941	. . . . . . . . . . . . . 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 2239	. . . . . . . . . . . . 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 2237	. . . . . . . . . . . 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 6081	. . . . . . . . . . . . 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 7700	. . . . . . . . . . . . 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 1249	. . . . . . . . . . . 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 4062	. . . . . . . . . 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 2615	. . . . . . . . 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 32	. . . . . . . 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 2615	. . . . . . 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 32	. . . . . 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 254	. . . . 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 7830	. . . . . . 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 4046	. . . . . 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 7832	. . . . . 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	bitrdi 196	. . . . 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 169	. . . 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	biimtrid 152	. . 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 6691	. 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵)))
120	6, 119	mpcom 36	1 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∃wrex 2476  ⟨cop 3626   class class class wbr 4034  (class class class)co 5925  [cec 6599  Pcnp 7375   +_P cpp 7377   ·_P cmp 7378  <_P cltp 7379   ~_R cer 7380  Rcnr 7381  0_Rc0r 7382   ·_R cmr 7386   <_R cltr 7387

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-i1p 7551  df-iplp 7552  df-imp 7553  df-iltp 7554  df-enr 7810  df-nr 7811  df-mr 7813  df-ltr 7814  df-0r 7815

This theorem is referenced by:  axpre-mulgt0  7971