Theorem xrsmulgzz 33189

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 7405	. . . 4 ⊢ (𝑛 = 0 → (𝑛(.g‘ℝ*𝑠)𝐵) = (0(.g‘ℝ*𝑠)𝐵))
2		oveq1 7405	. . . 4 ⊢ (𝑛 = 0 → (𝑛 ·e 𝐵) = (0 ·e 𝐵))
3	1, 2	eqeq12d 2780	. . 3 ⊢ (𝑛 = 0 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (0(.g‘ℝ*𝑠)𝐵) = (0 ·e 𝐵)))
4		oveq1 7405	. . . 4 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘ℝ*𝑠)𝐵) = (𝑚(.g‘ℝ*𝑠)𝐵))
5		oveq1 7405	. . . 4 ⊢ (𝑛 = 𝑚 → (𝑛 ·e 𝐵) = (𝑚 ·e 𝐵))
6	4, 5	eqeq12d 2780	. . 3 ⊢ (𝑛 = 𝑚 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)))
7		oveq1 7405	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (𝑛(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵))
8		oveq1 7405	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 ·e 𝐵) = ((𝑚 + 1) ·e 𝐵))
9	7, 8	eqeq12d 2780	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1) ·e 𝐵)))
10		oveq1 7405	. . . 4 ⊢ (𝑛 = -𝑚 → (𝑛(.g‘ℝ*𝑠)𝐵) = (-𝑚(.g‘ℝ*𝑠)𝐵))
11		oveq1 7405	. . . 4 ⊢ (𝑛 = -𝑚 → (𝑛 ·e 𝐵) = (-𝑚 ·e 𝐵))
12	10, 11	eqeq12d 2780	. . 3 ⊢ (𝑛 = -𝑚 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵)))
13		oveq1 7405	. . . 4 ⊢ (𝑛 = 𝐴 → (𝑛(.g‘ℝ*𝑠)𝐵) = (𝐴(.g‘ℝ*𝑠)𝐵))
14		oveq1 7405	. . . 4 ⊢ (𝑛 = 𝐴 → (𝑛 ·e 𝐵) = (𝐴 ·e 𝐵))
15	13, 14	eqeq12d 2780	. . 3 ⊢ (𝑛 = 𝐴 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)))
16		xrsbas 17638	. . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠)
17		xrs0 33186	. . . . 5 ⊢ 0 = (0g‘ℝ*𝑠)
18		eqid 2764	. . . . 5 ⊢ (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠)
19	16, 17, 18	mulg0 19118	. . . 4 ⊢ (𝐵 ∈ ℝ* → (0(.g‘ℝ*𝑠)𝐵) = 0)
20		xmul02 13273	. . . 4 ⊢ (𝐵 ∈ ℝ* → (0 ·e 𝐵) = 0)
21	19, 20	eqtr4d 2802	. . 3 ⊢ (𝐵 ∈ ℝ* → (0(.g‘ℝ*𝑠)𝐵) = (0 ·e 𝐵))
22		simpr 488	. . . . . 6 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵))
23	22	oveq1d 7413	. . . . 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 488	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
25		simpll 776	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ ℝ*)
26		xrsadd 21444	. . . . . . . . 9 ⊢ +𝑒 = (+g‘ℝ*𝑠)
27	16, 18, 26	mulgnnp1 19126	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
28	24, 25, 27	syl2anc 593	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
29		simpr 488	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → 𝑚 = 0)
30		simpll 776	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → 𝐵 ∈ ℝ*)
31		xaddlid 13247	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ* → (0 +𝑒 𝐵) = 𝐵)
32	31	adantl 485	. . . . . . . . 9 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (0 +𝑒 𝐵) = 𝐵)
33		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → 𝑚 = 0)
34	33	oveq1d 7413	. . . . . . . . . . 11 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) = (0(.g‘ℝ*𝑠)𝐵))
35	19	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (0(.g‘ℝ*𝑠)𝐵) = 0)
36	34, 35	eqtrd 2799	. . . . . . . . . 10 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) = 0)
37	36	oveq1d 7413	. . . . . . . . 9 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵) = (0 +𝑒 𝐵))
38	33	oveq1d 7413	. . . . . . . . . . . 12 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚 + 1) = (0 + 1))
39		0p1e1 12340	. . . . . . . . . . . 12 ⊢ (0 + 1) = 1
40	38, 39	eqtrdi 2815	. . . . . . . . . . 11 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚 + 1) = 1)
41	40	oveq1d 7413	. . . . . . . . . 10 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = (1(.g‘ℝ*𝑠)𝐵))
42	16, 18	mulg1 19125	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ* → (1(.g‘ℝ*𝑠)𝐵) = 𝐵)
43	42	adantl 485	. . . . . . . . . 10 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (1(.g‘ℝ*𝑠)𝐵) = 𝐵)
44	41, 43	eqtrd 2799	. . . . . . . . 9 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = 𝐵)
45	32, 37, 44	3eqtr4rd 2810	. . . . . . . 8 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
46	29, 30, 45	syl2anc 593	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
47		elnn0 12485	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 ↔ (𝑚 ∈ ℕ ∨ 𝑚 = 0))
48	47	bilani 508	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ ℕ ∨ 𝑚 = 0))
49	28, 46, 48	mpjaodan 971	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
50	49	adantr 484	. . . . 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))
51		nn0ssre 12487	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ
52		ressxr 11228	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
53	51, 52	sstri 3947	. . . . . . . 8 ⊢ ℕ0 ⊆ ℝ*
54		simpr 488	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
55	54	adantr 484	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℕ0)
56	53, 55	sselid 3936	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℝ*)
57		nn0ge0 12508	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → 0 ≤ 𝑚)
58	57	ad2antlr 737	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 0 ≤ 𝑚)
59		1xr 11243	. . . . . . . 8 ⊢ 1 ∈ ℝ*
60	59	a1i 11	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 1 ∈ ℝ*)
61		0le1 11712	. . . . . . . 8 ⊢ 0 ≤ 1
62	61	a1i 11	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 0 ≤ 1)
63		simpll 776	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝐵 ∈ ℝ*)
64		xadddi2r 13303	. . . . . . 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 1402	. . . . . 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	51, 55	sselid 3936	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℝ)
67		1re 11183	. . . . . . . . 9 ⊢ 1 ∈ ℝ
68	67	a1i 11	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 1 ∈ ℝ)
69		rexadd 13237	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑚 +𝑒 1) = (𝑚 + 1))
70	66, 68, 69	syl2anc 593	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (𝑚 +𝑒 1) = (𝑚 + 1))
71	70	oveq1d 7413	. . . . . 6 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 +𝑒 1) ·e 𝐵) = ((𝑚 + 1) ·e 𝐵))
72		xmullid 13285	. . . . . . . 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 7414	. . . . . 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 2807	. . . . 5 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 + 1) ·e 𝐵) = ((𝑚 ·e 𝐵) +𝑒 𝐵))
76	23, 50, 75	3eqtr4d 2809	. . . 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 423	. . 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 13212	. . . . . 6 ⊢ ((𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵) → -𝑒(𝑚(.g‘ℝ*𝑠)𝐵) = -𝑒(𝑚 ·e 𝐵))
79	78	adantl 485	. . . . 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 2764	. . . . . . . . 9 ⊢ (invg‘ℝ*𝑠) = (invg‘ℝ*𝑠)
81	16, 18, 80	mulgnegnn 19128	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (-𝑚(.g‘ℝ*𝑠)𝐵) = ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)))
82	81	ancoms 462	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑚(.g‘ℝ*𝑠)𝐵) = ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)))
83		xrsex 21443	. . . . . . . . . . . 12 ⊢ ℝ*𝑠 ∈ V
84	83	a1i 11	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ℝ*𝑠 ∈ V)
85		ssidd 3961	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ℝ* ⊆ ℝ*)
86		simp2 1151	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*)
87		simp3 1152	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*)
88	86, 87	xaddcld 13306	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 +𝑒 𝑦) ∈ ℝ*)
89	16, 18, 26, 84, 85, 88	mulgnnsubcl 19130	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
90	89	3anidm12 1440	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
91	90	ancoms 462	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
92		xrsinvgval 33188	. . . . . . . 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 2799	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑚(.g‘ℝ*𝑠)𝐵) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
95	94	adantr 484	. . . . 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 12219	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
97	96	adantl 485	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
98		rexneg 13216	. . . . . . . . 9 ⊢ (𝑚 ∈ ℝ → -𝑒𝑚 = -𝑚)
99	97, 98	syl 17	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → -𝑒𝑚 = -𝑚)
100	99	oveq1d 7413	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑒𝑚 ·e 𝐵) = (-𝑚 ·e 𝐵))
101		nnssre 12216	. . . . . . . . . 10 ⊢ ℕ ⊆ ℝ
102	101, 52	sstri 3947	. . . . . . . . 9 ⊢ ℕ ⊆ ℝ*
103		simpr 488	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
104	102, 103	sselid 3936	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ*)
105		simpl 486	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ ℝ*)
106		xmulneg1 13274	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
107	104, 105, 106	syl2anc 593	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑒𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
108	100, 107	eqtr3d 2801	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
109	108	adantr 484	. . . . 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 2809	. . . 4 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵))
111	110	exp31 423	. . 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 12676	. 2 ⊢ (𝐵 ∈ ℝ* → (𝐴 ∈ ℤ → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)))
113	112	impcom 411	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  Vcvv 3456   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078  ℝ^*cxr 11217   ≤ cle 11219  -cneg 11417  ℕcn 12212  ℕ₀cn0 12483  ℤcz 12570  -_𝑒cxne 13113   +_𝑒 cxad 13114   ·_e cxmu 13115  ℝ^*_𝑠cxrs 17532  inv_gcminusg 18978  ._gcmg 19111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-z 12571  df-dec 12691  df-uz 12842  df-xneg 13116  df-xadd 13117  df-xmul 13118  df-fz 13515  df-seq 14017  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17248  df-plusg 17301  df-mulr 17302  df-tset 17307  df-ple 17308  df-ds 17310  df-0g 17472  df-xrs 17534  df-minusg 18981  df-mulg 19112

This theorem is referenced by:  xrge0mulgnn0  33195  pnfinf  33365