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Theorem xmulasslem3 11988

Description: Lemma for xmulass 11989. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xmulasslem3	⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))

**Proof of Theorem xmulasslem3**
Step	Hyp	Ref	Expression
1		recn 9905	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
2		recn 9905	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
3		recn 9905	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ)
4		mulass 9903	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
5	1, 2, 3, 4	syl3an 1360	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
6	5	3expa 1257	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
7		remulcl 9900	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
8		rexmul 11973	. . . . . . . . 9 ⊢ (((𝐴 · 𝐵) ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) ·_e 𝐶) = ((𝐴 · 𝐵) · 𝐶))
9	7, 8	sylan 487	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) ·_e 𝐶) = ((𝐴 · 𝐵) · 𝐶))
10		remulcl 9900	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) ∈ ℝ)
11		rexmul 11973	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 · 𝐶) ∈ ℝ) → (𝐴 ·_e (𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶)))
12	10, 11	sylan2 490	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴 ·_e (𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶)))
13	12	anassrs 678	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐴 ·_e (𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶)))
14	6, 9, 13	3eqtr4d 2654	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 · 𝐶)))
15		rexmul 11973	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))
16	15	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))
17	16	oveq1d 6564	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = ((𝐴 · 𝐵) ·_e 𝐶))
18		rexmul 11973	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ·_e 𝐶) = (𝐵 · 𝐶))
19	18	adantll 746	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐵 ·_e 𝐶) = (𝐵 · 𝐶))
20	19	oveq2d 6565	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐴 ·_e (𝐵 ·_e 𝐶)) = (𝐴 ·_e (𝐵 · 𝐶)))
21	14, 17, 20	3eqtr4d 2654	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
22	21	adantll 746	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) ∧ 𝐶 ∈ ℝ) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
23		oveq2 6557	. . . . . . . . 9 ⊢ (𝐶 = +∞ → ((𝐴 ·_e 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐵) ·_e +∞))
24		simp1l 1078	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐴 ∈ ℝ^*)
25		simp2l 1080	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐵 ∈ ℝ^*)
26		xmulcl 11975	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)
27	24, 25, 26	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 ·_e 𝐵) ∈ ℝ^*)
28		xmulgt0 11985	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → 0 < (𝐴 ·_e 𝐵))
29	28	3adant3 1074	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < (𝐴 ·_e 𝐵))
30		xmulpnf1 11976	. . . . . . . . . 10 ⊢ (((𝐴 ·_e 𝐵) ∈ ℝ^* ∧ 0 < (𝐴 ·_e 𝐵)) → ((𝐴 ·_e 𝐵) ·_e +∞) = +∞)
31	27, 29, 30	syl2anc 691	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → ((𝐴 ·_e 𝐵) ·_e +∞) = +∞)
32	23, 31	sylan9eqr 2666	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = +∞)
33		simpl1 1057	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 ∈ ℝ^* ∧ 0 < 𝐴))
34		xmulpnf1 11976	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) → (𝐴 ·_e +∞) = +∞)
35	33, 34	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 ·_e +∞) = +∞)
36	32, 35	eqtr4d 2647	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e +∞))
37		oveq2 6557	. . . . . . . . 9 ⊢ (𝐶 = +∞ → (𝐵 ·_e 𝐶) = (𝐵 ·_e +∞))
38		xmulpnf1 11976	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (𝐵 ·_e +∞) = +∞)
39	38	3ad2ant2 1076	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐵 ·_e +∞) = +∞)
40	37, 39	sylan9eqr 2666	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐵 ·_e 𝐶) = +∞)
41	40	oveq2d 6565	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 ·_e (𝐵 ·_e 𝐶)) = (𝐴 ·_e +∞))
42	36, 41	eqtr4d 2647	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
43	42	adantlr 747	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) ∧ 𝐶 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
44		simpl3r 1110	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → 0 < 𝐶)
45		xmulasslem2 11984	. . . . . 6 ⊢ ((0 < 𝐶 ∧ 𝐶 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
46	44, 45	sylan 487	. . . . 5 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) ∧ 𝐶 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
47		simp3l 1082	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ^*)
48		elxr 11826	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* ↔ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
49	47, 48	sylib 207	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
50	49	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
51	22, 43, 46, 50	mpjao3dan 1387	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
52	51	anassrs 678	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
53		xmulpnf2 11977	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ^* ∧ 0 < 𝐶) → (+∞ ·_e 𝐶) = +∞)
54	53	3ad2ant3 1077	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (+∞ ·_e 𝐶) = +∞)
55	34	3ad2ant1 1075	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 ·_e +∞) = +∞)
56	54, 55	eqtr4d 2647	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (+∞ ·_e 𝐶) = (𝐴 ·_e +∞))
57	56	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → (+∞ ·_e 𝐶) = (𝐴 ·_e +∞))
58		oveq2 6557	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐴 ·_e 𝐵) = (𝐴 ·_e +∞))
59	58, 55	sylan9eqr 2666	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → (𝐴 ·_e 𝐵) = +∞)
60	59	oveq1d 6564	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (+∞ ·_e 𝐶))
61		oveq1 6556	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 ·_e 𝐶) = (+∞ ·_e 𝐶))
62	61, 54	sylan9eqr 2666	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → (𝐵 ·_e 𝐶) = +∞)
63	62	oveq2d 6565	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → (𝐴 ·_e (𝐵 ·_e 𝐶)) = (𝐴 ·_e +∞))
64	57, 60, 63	3eqtr4d 2654	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
65	64	adantlr 747	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
66		simpl2r 1108	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 ∈ ℝ) → 0 < 𝐵)
67		xmulasslem2 11984	. . . 4 ⊢ ((0 < 𝐵 ∧ 𝐵 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
68	66, 67	sylan 487	. . 3 ⊢ (((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 ∈ ℝ) ∧ 𝐵 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
69		elxr 11826	. . . . 5 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
70	25, 69	sylib 207	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
71	70	adantr 480	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 ∈ ℝ) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
72	52, 65, 68, 71	mpjao3dan 1387	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 ∈ ℝ) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
73		simpl3 1059	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → (𝐶 ∈ ℝ^* ∧ 0 < 𝐶))
74	73, 53	syl 17	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → (+∞ ·_e 𝐶) = +∞)
75		oveq1 6556	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐵) = (+∞ ·_e 𝐵))
76		xmulpnf2 11977	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (+∞ ·_e 𝐵) = +∞)
77	76	3ad2ant2 1076	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (+∞ ·_e 𝐵) = +∞)
78	75, 77	sylan9eqr 2666	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → (𝐴 ·_e 𝐵) = +∞)
79	78	oveq1d 6564	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (+∞ ·_e 𝐶))
80		oveq1 6556	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 ·_e (𝐵 ·_e 𝐶)) = (+∞ ·_e (𝐵 ·_e 𝐶)))
81		xmulcl 11975	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ·_e 𝐶) ∈ ℝ^)
82	25, 47, 81	syl2anc 691	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐵 ·_e 𝐶) ∈ ℝ^*)
83		xmulgt0 11985	. . . . . 6 ⊢ (((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < (𝐵 ·_e 𝐶))
84	83	3adant1 1072	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < (𝐵 ·_e 𝐶))
85		xmulpnf2 11977	. . . . 5 ⊢ (((𝐵 ·_e 𝐶) ∈ ℝ^* ∧ 0 < (𝐵 ·_e 𝐶)) → (+∞ ·_e (𝐵 ·_e 𝐶)) = +∞)
86	82, 84, 85	syl2anc 691	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (+∞ ·_e (𝐵 ·_e 𝐶)) = +∞)
87	80, 86	sylan9eqr 2666	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → (𝐴 ·_e (𝐵 ·_e 𝐶)) = +∞)
88	74, 79, 87	3eqtr4d 2654	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
89		simp1r 1079	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < 𝐴)
90		xmulasslem2 11984	. . 3 ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
91	89, 90	sylan 487	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
92		elxr 11826	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
93	24, 92	sylib 207	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
94	72, 88, 91, 93	mpjao3dan 1387	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 · cmul 9820 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ·_e cxmu 11821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-xmul 11824

This theorem is referenced by: xmulass 11989

W3C validator