Theorem xrsmulgzz 29806

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 6697	. . . 4 ⊢ (𝑛 = 0 → (𝑛(.g‘ℝ*𝑠)𝐵) = (0(.g‘ℝ*𝑠)𝐵))
2		oveq1 6697	. . . 4 ⊢ (𝑛 = 0 → (𝑛 ·e 𝐵) = (0 ·e 𝐵))
3	1, 2	eqeq12d 2666	. . 3 ⊢ (𝑛 = 0 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (0(.g‘ℝ*𝑠)𝐵) = (0 ·e 𝐵)))
4		oveq1 6697	. . . 4 ⊢ (𝑛 = 𝑚 → (𝑛(.g‘ℝ*𝑠)𝐵) = (𝑚(.g‘ℝ*𝑠)𝐵))
5		oveq1 6697	. . . 4 ⊢ (𝑛 = 𝑚 → (𝑛 ·e 𝐵) = (𝑚 ·e 𝐵))
6	4, 5	eqeq12d 2666	. . 3 ⊢ (𝑛 = 𝑚 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)))
7		oveq1 6697	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (𝑛(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵))
8		oveq1 6697	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 ·e 𝐵) = ((𝑚 + 1) ·e 𝐵))
9	7, 8	eqeq12d 2666	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1) ·e 𝐵)))
10		oveq1 6697	. . . 4 ⊢ (𝑛 = -𝑚 → (𝑛(.g‘ℝ*𝑠)𝐵) = (-𝑚(.g‘ℝ*𝑠)𝐵))
11		oveq1 6697	. . . 4 ⊢ (𝑛 = -𝑚 → (𝑛 ·e 𝐵) = (-𝑚 ·e 𝐵))
12	10, 11	eqeq12d 2666	. . 3 ⊢ (𝑛 = -𝑚 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵)))
13		oveq1 6697	. . . 4 ⊢ (𝑛 = 𝐴 → (𝑛(.g‘ℝ*𝑠)𝐵) = (𝐴(.g‘ℝ*𝑠)𝐵))
14		oveq1 6697	. . . 4 ⊢ (𝑛 = 𝐴 → (𝑛 ·e 𝐵) = (𝐴 ·e 𝐵))
15	13, 14	eqeq12d 2666	. . 3 ⊢ (𝑛 = 𝐴 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)))
16		xrsbas 19810	. . . . 5 ⊢ ℝ* = (Base‘ℝ*𝑠)
17		xrs0 29803	. . . . 5 ⊢ 0 = (0g‘ℝ*𝑠)
18		eqid 2651	. . . . 5 ⊢ (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠)
19	16, 17, 18	mulg0 17593	. . . 4 ⊢ (𝐵 ∈ ℝ* → (0(.g‘ℝ*𝑠)𝐵) = 0)
20		xmul02 12136	. . . 4 ⊢ (𝐵 ∈ ℝ* → (0 ·e 𝐵) = 0)
21	19, 20	eqtr4d 2688	. . 3 ⊢ (𝐵 ∈ ℝ* → (0(.g‘ℝ*𝑠)𝐵) = (0 ·e 𝐵))
22		simpr 476	. . . . . 6 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵))
23	22	oveq1d 6705	. . . . 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 476	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
25		simpll 805	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ ℝ*)
26		xrsadd 19811	. . . . . . . . 9 ⊢ +𝑒 = (+g‘ℝ*𝑠)
27	16, 18, 26	mulgnnp1 17596	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
28	24, 25, 27	syl2anc 694	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
29		simpr 476	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → 𝑚 = 0)
30		simpll 805	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → 𝐵 ∈ ℝ*)
31		xaddid2 12111	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ* → (0 +𝑒 𝐵) = 𝐵)
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (0 +𝑒 𝐵) = 𝐵)
33		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → 𝑚 = 0)
34	33	oveq1d 6705	. . . . . . . . . . 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 2685	. . . . . . . . . 10 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) = 0)
37	36	oveq1d 6705	. . . . . . . . 9 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵) = (0 +𝑒 𝐵))
38	33	oveq1d 6705	. . . . . . . . . . . 12 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚 + 1) = (0 + 1))
39		0p1e1 11170	. . . . . . . . . . . 12 ⊢ (0 + 1) = 1
40	38, 39	syl6eq 2701	. . . . . . . . . . 11 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚 + 1) = 1)
41	40	oveq1d 6705	. . . . . . . . . 10 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = (1(.g‘ℝ*𝑠)𝐵))
42	16, 18	mulg1 17595	. . . . . . . . . . 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 2685	. . . . . . . . 9 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = 𝐵)
45	32, 37, 44	3eqtr4rd 2696	. . . . . . . 8 ⊢ ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
46	29, 30, 45	syl2anc 694	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
47		simpr 476	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
48		elnn0 11332	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 ↔ (𝑚 ∈ ℕ ∨ 𝑚 = 0))
49	47, 48	sylib 208	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ ℕ ∨ 𝑚 = 0))
50	28, 46, 49	mpjaodan 844	. . . . . 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 11334	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ
53		ressxr 10121	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
54	52, 53	sstri 3645	. . . . . . . 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	sseldi 3634	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℝ*)
57		nn0ge0 11356	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → 0 ≤ 𝑚)
58	57	ad2antlr 763	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 0 ≤ 𝑚)
59		1re 10077	. . . . . . . . 9 ⊢ 1 ∈ ℝ
60	59	rexri 10135	. . . . . . . 8 ⊢ 1 ∈ ℝ*
61	60	a1i 11	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 1 ∈ ℝ*)
62		0le1 10589	. . . . . . . 8 ⊢ 0 ≤ 1
63	62	a1i 11	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 0 ≤ 1)
64		simpll 805	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝐵 ∈ ℝ*)
65		xadddi2r 12166	. . . . . . 7 ⊢ (((𝑚 ∈ ℝ* ∧ 0 ≤ 𝑚) ∧ (1 ∈ ℝ* ∧ 0 ≤ 1) ∧ 𝐵 ∈ ℝ*) → ((𝑚 +𝑒 1) ·e 𝐵) = ((𝑚 ·e 𝐵) +𝑒 (1 ·e 𝐵)))
66	56, 58, 61, 63, 64, 65	syl221anc 1377	. . . . . 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)))
67	52, 55	sseldi 3634	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℝ)
68	59	a1i 11	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 1 ∈ ℝ)
69		rexadd 12101	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑚 +𝑒 1) = (𝑚 + 1))
70	67, 68, 69	syl2anc 694	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (𝑚 +𝑒 1) = (𝑚 + 1))
71	70	oveq1d 6705	. . . . . 6 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 +𝑒 1) ·e 𝐵) = ((𝑚 + 1) ·e 𝐵))
72		xmulid2 12148	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ* → (1 ·e 𝐵) = 𝐵)
73	64, 72	syl 17	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (1 ·e 𝐵) = 𝐵)
74	73	oveq2d 6706	. . . . . 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	66, 71, 74	3eqtr3d 2693	. . . . 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 2695	. . . 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 629	. . 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 12076	. . . . . 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 2651	. . . . . . . . 9 ⊢ (invg‘ℝ*𝑠) = (invg‘ℝ*𝑠)
81	16, 18, 80	mulgnegnn 17598	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (-𝑚(.g‘ℝ*𝑠)𝐵) = ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)))
82	81	ancoms 468	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑚(.g‘ℝ*𝑠)𝐵) = ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)))
83		xrsex 19809	. . . . . . . . . . . 12 ⊢ ℝ*𝑠 ∈ V
84	83	a1i 11	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ℝ*𝑠 ∈ V)
85		ssid 3657	. . . . . . . . . . . 12 ⊢ ℝ* ⊆ ℝ*
86	85	a1i 11	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → ℝ* ⊆ ℝ*)
87		simp2 1082	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*)
88		simp3 1083	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*)
89	87, 88	xaddcld 12169	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥 +𝑒 𝑦) ∈ ℝ*)
90	16, 18, 26, 84, 86, 89	mulgnnsubcl 17600	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ∧ 𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
91	90	3anidm12 1423	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
92	91	ancoms 468	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
93		xrsinvgval 29805	. . . . . . . 8 ⊢ ((𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ* → ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
94	92, 93	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
95	82, 94	eqtrd 2685	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑚(.g‘ℝ*𝑠)𝐵) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
96	95	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))
97		nnre 11065	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
98	97	adantl 481	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
99		rexneg 12080	. . . . . . . . 9 ⊢ (𝑚 ∈ ℝ → -𝑒𝑚 = -𝑚)
100	98, 99	syl 17	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → -𝑒𝑚 = -𝑚)
101	100	oveq1d 6705	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑒𝑚 ·e 𝐵) = (-𝑚 ·e 𝐵))
102		nnssre 11062	. . . . . . . . . 10 ⊢ ℕ ⊆ ℝ
103	102, 53	sstri 3645	. . . . . . . . 9 ⊢ ℕ ⊆ ℝ*
104		simpr 476	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
105	103, 104	sseldi 3634	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ*)
106		simpl 472	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ ℝ*)
107		xmulneg1 12137	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
108	105, 106, 107	syl2anc 694	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑒𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
109	101, 108	eqtr3d 2687	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) → (-𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
110	109	adantr 480	. . . . 5 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (-𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
111	79, 96, 110	3eqtr4d 2695	. . . 4 ⊢ (((𝐵 ∈ ℝ* ∧ 𝑚 ∈ ℕ) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵))
112	111	exp31 629	. . 3 ⊢ (𝐵 ∈ ℝ* → (𝑚 ∈ ℕ → ((𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵) → (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵))))
113	3, 6, 9, 12, 15, 21, 77, 112	zindd 11516	. 2 ⊢ (𝐵 ∈ ℝ* → (𝐴 ∈ ℤ → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)))
114	113	impcom 445	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977  ℝ^*cxr 10111   ≤ cle 10113  -cneg 10305  ℕcn 11058  ℕ₀cn0 11330  ℤcz 11415  -_𝑒cxne 11981   +_𝑒 cxad 11982   ·_e cxmu 11983  ℝ^*_𝑠cxrs 16207  inv_gcminusg 17470  ._gcmg 17587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-fz 12365  df-seq 12842  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-plusg 16001  df-mulr 16002  df-tset 16007  df-ple 16008  df-ds 16011  df-0g 16149  df-xrs 16209  df-minusg 17473  df-mulg 17588

This theorem is referenced by:  xrge0mulgnn0  29817  pnfinf  29865