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Theorem xmulasslem3 13307

Description: Lemma for xmulass 13308. (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 11238	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
2		recn 11238	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
3		recn 11238	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ)
4		mulass 11236	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
5	1, 2, 3, 4	syl3an 1157	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
6	5	3expa 1115	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
7		remulcl 11233	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
8		rexmul 13292	. . . . . . . . 9 ⊢ (((𝐴 · 𝐵) ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) ·_e 𝐶) = ((𝐴 · 𝐵) · 𝐶))
9	7, 8	sylan 578	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) ·_e 𝐶) = ((𝐴 · 𝐵) · 𝐶))
10		remulcl 11233	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) ∈ ℝ)
11		rexmul 13292	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 · 𝐶) ∈ ℝ) → (𝐴 ·_e (𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶)))
12	10, 11	sylan2 591	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴 ·_e (𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶)))
13	12	anassrs 466	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐴 ·_e (𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶)))
14	6, 9, 13	3eqtr4d 2778	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 · 𝐶)))
15		rexmul 13292	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))
16	15	adantr 479	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))
17	16	oveq1d 7441	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = ((𝐴 · 𝐵) ·_e 𝐶))
18		rexmul 13292	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ·_e 𝐶) = (𝐵 · 𝐶))
19	18	adantll 712	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐵 ·_e 𝐶) = (𝐵 · 𝐶))
20	19	oveq2d 7442	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐴 ·_e (𝐵 ·_e 𝐶)) = (𝐴 ·_e (𝐵 · 𝐶)))
21	14, 17, 20	3eqtr4d 2778	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
22	21	adantll 712	. . . . 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 7434	. . . . . . . . 9 ⊢ (𝐶 = +∞ → ((𝐴 ·_e 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐵) ·_e +∞))
24		simp1l 1194	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐴 ∈ ℝ^*)
25		simp2l 1196	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐵 ∈ ℝ^*)
26		xmulcl 13294	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)
27	24, 25, 26	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 ·_e 𝐵) ∈ ℝ^*)
28		xmulgt0 13304	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵)) → 0 < (𝐴 ·_e 𝐵))
29	28	3adant3 1129	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < (𝐴 ·_e 𝐵))
30		xmulpnf1 13295	. . . . . . . . . 10 ⊢ (((𝐴 ·_e 𝐵) ∈ ℝ^* ∧ 0 < (𝐴 ·_e 𝐵)) → ((𝐴 ·_e 𝐵) ·_e +∞) = +∞)
31	27, 29, 30	syl2anc 582	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → ((𝐴 ·_e 𝐵) ·_e +∞) = +∞)
32	23, 31	sylan9eqr 2790	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = +∞)
33		simpl1 1188	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐴 ∈ ℝ^* ∧ 0 < 𝐴))
34		xmulpnf1 13295	. . . . . . . . 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 2771	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e +∞))
37		oveq2 7434	. . . . . . . . 9 ⊢ (𝐶 = +∞ → (𝐵 ·_e 𝐶) = (𝐵 ·_e +∞))
38		xmulpnf1 13295	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (𝐵 ·_e +∞) = +∞)
39	38	3ad2ant2 1131	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐵 ·_e +∞) = +∞)
40	37, 39	sylan9eqr 2790	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐵 ·_e 𝐶) = +∞)
41	40	oveq2d 7442	. . . . . . 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 2771	. . . . . 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 713	. . . . 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 1226	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → 0 < 𝐶)
45		xmulasslem2 13303	. . . . . 6 ⊢ ((0 < 𝐶 ∧ 𝐶 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
46	44, 45	sylan 578	. . . . 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 1198	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ^*)
48		elxr 13138	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* ↔ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
49	47, 48	sylib 217	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
50	49	adantr 479	. . . . 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 1428	. . . 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 466	. . 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 13296	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ^* ∧ 0 < 𝐶) → (+∞ ·_e 𝐶) = +∞)
54	53	3ad2ant3 1132	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (+∞ ·_e 𝐶) = +∞)
55	34	3ad2ant1 1130	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 ·_e +∞) = +∞)
56	54, 55	eqtr4d 2771	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (+∞ ·_e 𝐶) = (𝐴 ·_e +∞))
57	56	adantr 479	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → (+∞ ·_e 𝐶) = (𝐴 ·_e +∞))
58		oveq2 7434	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐴 ·_e 𝐵) = (𝐴 ·_e +∞))
59	58, 55	sylan9eqr 2790	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → (𝐴 ·_e 𝐵) = +∞)
60	59	oveq1d 7441	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (+∞ ·_e 𝐶))
61		oveq1 7433	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 ·_e 𝐶) = (+∞ ·_e 𝐶))
62	61, 54	sylan9eqr 2790	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → (𝐵 ·_e 𝐶) = +∞)
63	62	oveq2d 7442	. . . . 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 2778	. . . 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 713	. . 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 1224	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 ∈ ℝ) → 0 < 𝐵)
67		xmulasslem2 13303	. . . 4 ⊢ ((0 < 𝐵 ∧ 𝐵 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
68	66, 67	sylan 578	. . 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 13138	. . . . 5 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
70	25, 69	sylib 217	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
71	70	adantr 479	. . 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 1428	. 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 1190	. . . 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 7433	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐵) = (+∞ ·_e 𝐵))
76		xmulpnf2 13296	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) → (+∞ ·_e 𝐵) = +∞)
77	76	3ad2ant2 1131	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (+∞ ·_e 𝐵) = +∞)
78	75, 77	sylan9eqr 2790	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → (𝐴 ·_e 𝐵) = +∞)
79	78	oveq1d 7441	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (+∞ ·_e 𝐶))
80		oveq1 7433	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 ·_e (𝐵 ·_e 𝐶)) = (+∞ ·_e (𝐵 ·_e 𝐶)))
81		xmulcl 13294	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ·_e 𝐶) ∈ ℝ^)
82	25, 47, 81	syl2anc 582	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐵 ·_e 𝐶) ∈ ℝ^*)
83		xmulgt0 13304	. . . . . 6 ⊢ (((𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < (𝐵 ·_e 𝐶))
84	83	3adant1 1127	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < (𝐵 ·_e 𝐶))
85		xmulpnf2 13296	. . . . 5 ⊢ (((𝐵 ·_e 𝐶) ∈ ℝ^* ∧ 0 < (𝐵 ·_e 𝐶)) → (+∞ ·_e (𝐵 ·_e 𝐶)) = +∞)
86	82, 84, 85	syl2anc 582	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (+∞ ·_e (𝐵 ·_e 𝐶)) = +∞)
87	80, 86	sylan9eqr 2790	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → (𝐴 ·_e (𝐵 ·_e 𝐶)) = +∞)
88	74, 79, 87	3eqtr4d 2778	. 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 1195	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → 0 < 𝐴)
90		xmulasslem2 13303	. . 3 ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
91	89, 90	sylan 578	. 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 13138	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
93	24, 92	sylib 217	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
94	72, 88, 91, 93	mpjao3dan 1428	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 394 ∨ w3o 1083 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7426 ℂcc 11146 ℝcr 11147 0cc0 11148 · cmul 11153 +∞cpnf 11285 -∞cmnf 11286 ℝ^*cxr 11287 < clt 11288 ·_e cxmu 13133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-xmul 13136

This theorem is referenced by: xmulass 13308

W3C validator