Theorem mulgt0sr 8109

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 8069	. . . . 5 ⊢ <R ⊆ (R × R)
2	1	brel 4807	. . . 4 ⊢ (0R <R 𝐴 → (0R ∈ R ∧ 𝐴 ∈ R))
3	2	simprd 114	. . 3 ⊢ (0R <R 𝐴 → 𝐴 ∈ R)
4	1	brel 4807	. . . 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 8058	. . 3 ⊢ R = ((P × P) / ~R )
8		breq2 4118	. . . . 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 6065	. . . . 5 ⊢ ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ))
11	10	breq2d 4126	. . . 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 4118	. . . . 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 6066	. . . . 5 ⊢ ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 ·R [⟨𝑧, 𝑤⟩] ~R ) = (𝐴 ·R 𝐵))
16	15	breq2d 4126	. . . 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 8079	. . . . 5 ⊢ (0R <R [⟨𝑥, 𝑦⟩] ~R ↔ 𝑦<P 𝑥)
19		gt0srpr 8079	. . . . 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 7944	. . . . . . 7 ⊢ (𝑦<P 𝑥 → ∃𝑣 ∈ P (𝑦 +P 𝑣) = 𝑥)
22		ltexpri 7944	. . . . . . . . 9 ⊢ (𝑤<P 𝑧 → ∃𝑢 ∈ P (𝑤 +P 𝑢) = 𝑧)
23		addclpr 7868	. . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . . 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 6216	. . . . . . . . . . . . . . 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 2312	. . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . 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 7903	. . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . 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 7903	. . . . . . . . . . . . . 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 6216	. . . . . . . . . . . 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 531	. . . . . . . . . . . . 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 7903	. . . . . . . . . . . . 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 7928	. . . . . . . . . . . 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 533	. . . . . . . . . . . . . . 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 6067	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 +P 𝑣) = 𝑥 ∧ (𝑤 +P 𝑢) = 𝑧) → ((𝑦 +P 𝑣) ·P (𝑤 +P 𝑢)) = (𝑥 ·P 𝑧))
47	46	oveq1d 6073	. . . . . . . . . . . . . . 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 7919	. . . . . . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . . . . . 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 6066	. . . . . . . . . . . . . . . . . . . 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 2269	. . . . . . . . . . . . . . . . 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 6073	. . . . . . . . . . . . . . . 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 7919	. . . . . . . . . . . . . . . . . . . 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 7911	. . . . . . . . . . . . . . . . . . . 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 6239	. . . . . . . . . . . . . . . . . 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 6239	. . . . . . . . . . . . . . . . . 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 6076	. . . . . . . . . . . . . . . . 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 7919	. . . . . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . . . . 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 7903	. . . . . . . . . . . . . . . . . . 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 7903	. . . . . . . . . . . . . . . . . . 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 7903	. . . . . . . . . . . . . . . . . . 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 7909	. . . . . . . . . . . . . . . . . . 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 7910	. . . . . . . . . . . . . . . . . . 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 6247	. . . . . . . . . . . . . . . . 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 2277	. . . . . . . . . . . . . . . 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 6244	. . . . . . . . . . . . . . . 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 2277	. . . . . . . . . . . . . . 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 6065	. . . . . . . . . . . . . . . . . 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 2269	. . . . . . . . . . . . . . 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 6076	. . . . . . . . . . . . . 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 2269	. . . . . . . . . . . . 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 7903	. . . . . . . . . . . . . . . 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 7910	. . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . 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 7868	. . . . . . . . . . . . . . . 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 7909	. . . . . . . . . . . . . . 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 2269	. . . . . . . . . . . . 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 6216	. . . . . . . . . . . . . . 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 7910	. . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . 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 6243	. . . . . . . . . . . . . 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 7910	. . . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . . 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 6074	. . . . . . . . . . . . . 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 2277	. . . . . . . . . . . . 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 2275	. . . . . . . . . . . 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 6216	. . . . . . . . . . . . 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 7947	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 4142	. . . . . . . . . 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 2663	. . . . . . . . 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 2663	. . . . . . 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 8077	. . . . . . 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 4126	. . . . . 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 8079	. . . . . 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 6870	. 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 1005   = wceq 1398   ∈ wcel 2205  ∃wrex 2523  ⟨cop 3697   class class class wbr 4114  (class class class)co 6058  [cec 6778  Pcnp 7622   +_P cpp 7624   ·_P cmp 7625  <_P cltp 7626   ~_R cer 7627  Rcnr 7628  0_Rc0r 7629   ·_R cmr 7633   <_R cltr 7634

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-i1p 7798  df-iplp 7799  df-imp 7800  df-iltp 7801  df-enr 8057  df-nr 8058  df-mr 8060  df-ltr 8061  df-0r 8062

This theorem is referenced by:  axpre-mulgt0  8218