Theorem xrsmulgzz 33070

Description: The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)

Assertion

Ref	Expression
xrsmulgzz	⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵))

Proof of Theorem xrsmulgzz

Dummy variables 𝑛 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7365	. . . 4 ⊢ (𝑛 = 0 → (𝑛(.g‘ℝ*𝑠)𝐵) = (0(.g‘ℝ*𝑠)𝐵))
2		oveq1 7365	. . . 4 ⊢ (𝑛 = 0 → (𝑛 ·e 𝐵) = (0 ·e 𝐵))
3	1, 2	eqeq12d 2751	. . 3 ⊢ (𝑛 = 0 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (0(.g‘ℝ*𝑠)𝐵) = (0 ·e 𝐵)))
4		oveq1 7365	. . . 4 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘ℝ*𝑠)𝐵) = (𝑚(.g‘ℝ*𝑠)𝐵))
5		oveq1 7365	. . . 4 ⊢ (𝑛 = 𝑚 → (𝑛 ·e 𝐵) = (𝑚 ·e 𝐵))
6	4, 5	eqeq12d 2751	. . 3 ⊢ (𝑛 = 𝑚 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)))
7		oveq1 7365	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (𝑛(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵))
8		oveq1 7365	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 ·e 𝐵) = ((𝑚 + 1) ·e 𝐵))
9	7, 8	eqeq12d 2751	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1) ·e 𝐵)))
10		oveq1 7365	. . . 4 ⊢ (𝑛 = -𝑚 → (𝑛(.g‘ℝ*𝑠)𝐵) = (-𝑚(.g‘ℝ*𝑠)𝐵))
11		oveq1 7365	. . . 4 ⊢ (𝑛 = -𝑚 → (𝑛 ·e 𝐵) = (-𝑚 ·e 𝐵))
12	10, 11	eqeq12d 2751	. . 3 ⊢ (𝑛 = -𝑚 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵)))
13		oveq1 7365	. . . 4 ⊢ (𝑛 = 𝐴 → (𝑛(.g‘ℝ*𝑠)𝐵) = (𝐴(.g‘ℝ*𝑠)𝐵))
14		oveq1 7365	. . . 4 ⊢ (𝑛 = 𝐴 → (𝑛 ·e 𝐵) = (𝐴 ·e 𝐵))
15	13, 14	eqeq12d 2751	. . 3 ⊢ (𝑛 = 𝐴 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)))
16		xrsbas 17529	. . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠)
17		xrs0 33067	. . . . 5 ⊢ 0 = (0g‘ℝ*𝑠)
18		eqid 2735	. . . . 5 ⊢ (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠)
19	16, 17, 18	mulg0 19006	. . . 4 ⊢ (𝐵 ∈ ℝ* → (0(.g‘ℝ*𝑠)𝐵) = 0)
20		xmul02 13185	. . . 4 ⊢ (𝐵 ∈ ℝ* → (0 ·e 𝐵) = 0)
21	19, 20	eqtr4d 2773	. . 3 ⊢ (𝐵 ∈ ℝ* → (0(.g‘ℝ*𝑠)𝐵) = (0 ·e 𝐵))
22		simpr 484	. . . . . 6 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵))
23	22	oveq1d 7373	. . . . 5 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵) = ((𝑚 ·e 𝐵) +𝑒 𝐵))
24		simpr 484	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
25		simpll 767	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ ℝ*)
26		xrsadd 21342	. . . . . . . . 9 ⊢ +𝑒 = (+g‘ℝ*𝑠)
27	16, 18, 26	mulgnnp1 19014	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
28	24, 25, 27	syl2anc 585	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
29		simpr 484	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → 𝑚 = 0)
30		simpll 767	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → 𝐵 ∈ ℝ*)
31		xaddlid 13159	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ* → (0 +𝑒 𝐵) = 𝐵)
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (0 +𝑒 𝐵) = 𝐵)
33		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → 𝑚 = 0)
34	33	oveq1d 7373	. . . . . . . . . . 11 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) = (0(.g‘ℝ*𝑠)𝐵))
35	19	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (0(.g‘ℝ*𝑠)𝐵) = 0)
36	34, 35	eqtrd 2770	. . . . . . . . . 10 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) = 0)
37	36	oveq1d 7373	. . . . . . . . 9 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵) = (0 +𝑒 𝐵))
38	33	oveq1d 7373	. . . . . . . . . . . 12 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚 + 1) = (0 + 1))
39		0p1e1 12264	. . . . . . . . . . . 12 ⊢ (0 + 1) = 1
40	38, 39	eqtrdi 2786	. . . . . . . . . . 11 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚 + 1) = 1)
41	40	oveq1d 7373	. . . . . . . . . 10 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = (1(.g‘ℝ*𝑠)𝐵))
42	16, 18	mulg1 19013	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ* → (1(.g‘ℝ*𝑠)𝐵) = 𝐵)
43	42	adantl 481	. . . . . . . . . 10 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (1(.g‘ℝ*𝑠)𝐵) = 𝐵)
44	41, 43	eqtrd 2770	. . . . . . . . 9 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = 𝐵)
45	32, 37, 44	3eqtr4rd 2781	. . . . . . . 8 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
46	29, 30, 45	syl2anc 585	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
47		simpr 484	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
48		elnn0 12405	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 ↔ (𝑚 ∈ ℕ ∨ 𝑚 = 0))
49	47, 48	sylib 218	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ ℕ ∨ 𝑚 = 0))
50	28, 46, 49	mpjaodan 961	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
51	50	adantr 480	. . . . 5 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
52		nn0ssre 12407	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ
53		ressxr 11178	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
54	52, 53	sstri 3942	. . . . . . . 8 ⊢ ℕ0 ⊆ ℝ*
55	47	adantr 480	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℕ0)
56	54, 55	sselid 3930	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℝ*)
57		nn0ge0 12428	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → 0 ≤ 𝑚)
58	57	ad2antlr 728	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 0 ≤ 𝑚)
59		1xr 11193	. . . . . . . 8 ⊢ 1 ∈ ℝ*
60	59	a1i 11	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 1 ∈ ℝ*)
61		0le1 11662	. . . . . . . 8 ⊢ 0 ≤ 1
62	61	a1i 11	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 0 ≤ 1)
63		simpll 767	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝐵 ∈ ℝ*)
64		xadddi2r 13215	. . . . . . 7 ⊢ (((𝑚 ∈ ℝ* ∧ 0 ≤ 𝑚) ∧ (1 ∈ ℝ* ∧ 0 ≤ 1) ∧ 𝐵 ∈ ℝ*) → ((𝑚 +𝑒 1) ·e 𝐵) = ((𝑚 ·e 𝐵) +𝑒 (1 ·e 𝐵)))
65	56, 58, 60, 62, 63, 64	syl221anc 1384	. . . . . 6 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 +𝑒 1) ·e 𝐵) = ((𝑚 ·e 𝐵) +𝑒 (1 ·e 𝐵)))
66	52, 55	sselid 3930	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℝ)
67		1re 11134	. . . . . . . . 9 ⊢ 1 ∈ ℝ
68	67	a1i 11	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 1 ∈ ℝ)
69		rexadd 13149	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑚 +𝑒 1) = (𝑚 + 1))
70	66, 68, 69	syl2anc 585	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (𝑚 +𝑒 1) = (𝑚 + 1))
71	70	oveq1d 7373	. . . . . 6 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 +𝑒 1) ·e 𝐵) = ((𝑚 + 1) ·e 𝐵))
72		xmullid 13197	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ* → (1 ·e 𝐵) = 𝐵)
73	63, 72	syl 17	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (1 ·e 𝐵) = 𝐵)
74	73	oveq2d 7374	. . . . . 6 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 ·e 𝐵) +𝑒 (1 ·e 𝐵)) = ((𝑚 ·e 𝐵) +𝑒 𝐵))
75	65, 71, 74	3eqtr3d 2778	. . . . 5 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 + 1) ·e 𝐵) = ((𝑚 ·e 𝐵) +𝑒 𝐵))
76	23, 51, 75	3eqtr4d 2780	. . . 4 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1) ·e 𝐵))
77	76	exp31 419	. . 3 ⊢ (𝐵 ∈ ℝ* → (𝑚 ∈ ℕ0 → ((𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1) ·e 𝐵))))
78		xnegeq 13124	. . . . . 6 ⊢ ((𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵) → -𝑒(𝑚(.g‘ℝ*𝑠)𝐵) = -𝑒(𝑚 ·e 𝐵))
79	78	adantl 481	. . . . 5 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → -𝑒(𝑚(.g‘ℝ*𝑠)𝐵) = -𝑒(𝑚 ·e 𝐵))
80		eqid 2735	. . . . . . . . 9 ⊢ (invg‘ℝ*𝑠) = (invg‘ℝ*𝑠)
81	16, 18, 80	mulgnegnn 19016	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (-𝑚(.g‘ℝ*𝑠)𝐵) = ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)))
82	81	ancoms 458	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑚(.g‘ℝ*𝑠)𝐵) = ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)))
83		xrsex 21341	. . . . . . . . . . . 12 ⊢ ℝ*𝑠 ∈ V
84	83	a1i 11	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ℝ*𝑠 ∈ V)
85		ssidd 3956	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ℝ* ⊆ ℝ*)
86		simp2 1138	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*)
87		simp3 1139	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*)
88	86, 87	xaddcld 13218	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 +𝑒 𝑦) ∈ ℝ*)
89	16, 18, 26, 84, 85, 88	mulgnnsubcl 19018	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
90	89	3anidm12 1422	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
91	90	ancoms 458	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
92		xrsinvgval 33069	. . . . . . . 8 ⊢ ((𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ* → ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
93	91, 92	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
94	82, 93	eqtrd 2770	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑚(.g‘ℝ*𝑠)𝐵) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
95	94	adantr 480	. . . . 5 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (-𝑚(.g‘ℝ*𝑠)𝐵) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
96		nnre 12154	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
97	96	adantl 481	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
98		rexneg 13128	. . . . . . . . 9 ⊢ (𝑚 ∈ ℝ → -𝑒𝑚 = -𝑚)
99	97, 98	syl 17	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → -𝑒𝑚 = -𝑚)
100	99	oveq1d 7373	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑒𝑚 ·e 𝐵) = (-𝑚 ·e 𝐵))
101		nnssre 12151	. . . . . . . . . 10 ⊢ ℕ ⊆ ℝ
102	101, 53	sstri 3942	. . . . . . . . 9 ⊢ ℕ ⊆ ℝ*
103		simpr 484	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
104	102, 103	sselid 3930	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ*)
105		simpl 482	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ ℝ*)
106		xmulneg1 13186	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
107	104, 105, 106	syl2anc 585	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑒𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
108	100, 107	eqtr3d 2772	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
109	108	adantr 480	. . . . 5 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (-𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
110	79, 95, 109	3eqtr4d 2780	. . . 4 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵))
111	110	exp31 419	. . 3 ⊢ (𝐵 ∈ ℝ* → (𝑚 ∈ ℕ → ((𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵) → (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵))))
112	3, 6, 9, 12, 15, 21, 77, 111	zindd 12595	. 2 ⊢ (𝐵 ∈ ℝ* → (𝐴 ∈ ℤ → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)))
113	112	impcom 407	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3439   class class class wbr 5097  ‘cfv 6491  (class class class)co 7358  ℝcr 11027  0cc0 11028  1c1 11029   + caddc 11031  ℝ^*cxr 11167   ≤ cle 11169  -cneg 11367  ℕcn 12147  ℕ₀cn0 12403  ℤcz 12490  -_𝑒cxne 13025   +_𝑒 cxad 13026   ·_e cxmu 13027  ℝ^*_𝑠cxrs 17423  inv_gcminusg 18866  ._gcmg 18999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12610  df-uz 12754  df-xneg 13028  df-xadd 13029  df-xmul 13030  df-fz 13426  df-seq 13927  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17139  df-plusg 17192  df-mulr 17193  df-tset 17198  df-ple 17199  df-ds 17201  df-0g 17363  df-xrs 17425  df-minusg 18869  df-mulg 19000

This theorem is referenced by:  xrge0mulgnn0  33076  pnfinf  33244