xmulasslem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xmulasslem3		Structured version Visualization version GIF version

Theorem xmulasslem3 13271

Description: Lemma for xmulass 13272. (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 11202	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
2		recn 11202	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
3		recn 11202	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ)
4		mulass 11200	. . . . . . . . . 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 11197	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
8		rexmul 13256	. . . . . . . . 9 ⊢ (((𝐴 · 𝐵) ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) ·_e 𝐶) = ((𝐴 · 𝐵) · 𝐶))
9	7, 8	sylan 579	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) ·_e 𝐶) = ((𝐴 · 𝐵) · 𝐶))
10		remulcl 11197	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) ∈ ℝ)
11		rexmul 13256	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 · 𝐶) ∈ ℝ) → (𝐴 ·_e (𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶)))
12	10, 11	sylan2 592	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) → (𝐴 ·_e (𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶)))
13	12	anassrs 467	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐴 ·_e (𝐵 · 𝐶)) = (𝐴 · (𝐵 · 𝐶)))
14	6, 9, 13	3eqtr4d 2776	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 · 𝐶)))
15		rexmul 13256	. . . . . . . . 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 7420	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = ((𝐴 · 𝐵) ·_e 𝐶))
18		rexmul 13256	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 ·_e 𝐶) = (𝐵 · 𝐶))
19	18	adantll 711	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐵 ·_e 𝐶) = (𝐵 · 𝐶))
20	19	oveq2d 7421	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → (𝐴 ·_e (𝐵 ·_e 𝐶)) = (𝐴 ·_e (𝐵 · 𝐶)))
21	14, 17, 20	3eqtr4d 2776	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
22	21	adantll 711	. . . . 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 7413	. . . . . . . . 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 13258	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)
27	24, 25, 26	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐴 ·_e 𝐵) ∈ ℝ^*)
28		xmulgt0 13268	. . . . . . . . . . 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 13259	. . . . . . . . . 10 ⊢ (((𝐴 ·_e 𝐵) ∈ ℝ^* ∧ 0 < (𝐴 ·_e 𝐵)) → ((𝐴 ·_e 𝐵) ·_e +∞) = +∞)
31	27, 29, 30	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → ((𝐴 ·_e 𝐵) ·_e +∞) = +∞)
32	23, 31	sylan9eqr 2788	. . . . . . . 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 13259	. . . . . . . . 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 2769	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e +∞))
37		oveq2 7413	. . . . . . . . 9 ⊢ (𝐶 = +∞ → (𝐵 ·_e 𝐶) = (𝐵 ·_e +∞))
38		xmulpnf1 13259	. . . . . . . . . 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 2788	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐶 = +∞) → (𝐵 ·_e 𝐶) = +∞)
41	40	oveq2d 7421	. . . . . . 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 2769	. . . . . 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 712	. . . . 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 13267	. . . . . 6 ⊢ ((0 < 𝐶 ∧ 𝐶 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
46	44, 45	sylan 579	. . . . 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 13102	. . . . . . 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 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 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 467	. . 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 13260	. . . . . . . 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 2769	. . . . . 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 7413	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐴 ·_e 𝐵) = (𝐴 ·_e +∞))
59	58, 55	sylan9eqr 2788	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → (𝐴 ·_e 𝐵) = +∞)
60	59	oveq1d 7420	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (+∞ ·_e 𝐶))
61		oveq1 7412	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 ·_e 𝐶) = (+∞ ·_e 𝐶))
62	61, 54	sylan9eqr 2788	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐵 = +∞) → (𝐵 ·_e 𝐶) = +∞)
63	62	oveq2d 7421	. . . . 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 2776	. . . 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 712	. . 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 13267	. . . 4 ⊢ ((0 < 𝐵 ∧ 𝐵 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
68	66, 67	sylan 579	. . 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 13102	. . . . 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 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 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 7412	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 ·_e 𝐵) = (+∞ ·_e 𝐵))
76		xmulpnf2 13260	. . . . . 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 2788	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → (𝐴 ·_e 𝐵) = +∞)
79	78	oveq1d 7420	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (+∞ ·_e 𝐶))
80		oveq1 7412	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 ·_e (𝐵 ·_e 𝐶)) = (+∞ ·_e (𝐵 ·_e 𝐶)))
81		xmulcl 13258	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ·_e 𝐶) ∈ ℝ^)
82	25, 47, 81	syl2anc 583	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (𝐵 ·_e 𝐶) ∈ ℝ^*)
83		xmulgt0 13268	. . . . . 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 13260	. . . . 5 ⊢ (((𝐵 ·_e 𝐶) ∈ ℝ^* ∧ 0 < (𝐵 ·_e 𝐶)) → (+∞ ·_e (𝐵 ·_e 𝐶)) = +∞)
86	82, 84, 85	syl2anc 583	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) → (+∞ ·_e (𝐵 ·_e 𝐶)) = +∞)
87	80, 86	sylan9eqr 2788	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ^* ∧ 0 < 𝐶)) ∧ 𝐴 = +∞) → (𝐴 ·_e (𝐵 ·_e 𝐶)) = +∞)
88	74, 79, 87	3eqtr4d 2776	. 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 13267	. . 3 ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → ((𝐴 ·_e 𝐵) ·_e 𝐶) = (𝐴 ·_e (𝐵 ·_e 𝐶)))
91	89, 90	sylan 579	. 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 13102	. . 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 395 ∨ w3o 1083 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7405 ℂcc 11110 ℝcr 11111 0cc0 11112 · cmul 11117 +∞cpnf 11249 -∞cmnf 11250 ℝ^*cxr 11251 < clt 11252 ·_e cxmu 13097

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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-xmul 13100

This theorem is referenced by: xmulass 13272

W3C validator