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Theorem xrsmulgzz 29009
 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.
StepHypRef Expression
1 oveq1 6556 . . . 4 (𝑛 = 0 → (𝑛(.g‘ℝ*𝑠)𝐵) = (0(.g‘ℝ*𝑠)𝐵))
2 oveq1 6556 . . . 4 (𝑛 = 0 → (𝑛 ·e 𝐵) = (0 ·e 𝐵))
31, 2eqeq12d 2625 . . 3 (𝑛 = 0 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (0(.g‘ℝ*𝑠)𝐵) = (0 ·e 𝐵)))
4 oveq1 6556 . . . 4 (𝑛 = 𝑚 → (𝑛(.g‘ℝ*𝑠)𝐵) = (𝑚(.g‘ℝ*𝑠)𝐵))
5 oveq1 6556 . . . 4 (𝑛 = 𝑚 → (𝑛 ·e 𝐵) = (𝑚 ·e 𝐵))
64, 5eqeq12d 2625 . . 3 (𝑛 = 𝑚 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)))
7 oveq1 6556 . . . 4 (𝑛 = (𝑚 + 1) → (𝑛(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵))
8 oveq1 6556 . . . 4 (𝑛 = (𝑚 + 1) → (𝑛 ·e 𝐵) = ((𝑚 + 1) ·e 𝐵))
97, 8eqeq12d 2625 . . 3 (𝑛 = (𝑚 + 1) → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1) ·e 𝐵)))
10 oveq1 6556 . . . 4 (𝑛 = -𝑚 → (𝑛(.g‘ℝ*𝑠)𝐵) = (-𝑚(.g‘ℝ*𝑠)𝐵))
11 oveq1 6556 . . . 4 (𝑛 = -𝑚 → (𝑛 ·e 𝐵) = (-𝑚 ·e 𝐵))
1210, 11eqeq12d 2625 . . 3 (𝑛 = -𝑚 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵)))
13 oveq1 6556 . . . 4 (𝑛 = 𝐴 → (𝑛(.g‘ℝ*𝑠)𝐵) = (𝐴(.g‘ℝ*𝑠)𝐵))
14 oveq1 6556 . . . 4 (𝑛 = 𝐴 → (𝑛 ·e 𝐵) = (𝐴 ·e 𝐵))
1513, 14eqeq12d 2625 . . 3 (𝑛 = 𝐴 → ((𝑛(.g‘ℝ*𝑠)𝐵) = (𝑛 ·e 𝐵) ↔ (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)))
16 xrsbas 19581 . . . . 5 * = (Base‘ℝ*𝑠)
17 xrs0 29006 . . . . 5 0 = (0g‘ℝ*𝑠)
18 eqid 2610 . . . . 5 (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠)
1916, 17, 18mulg0 17369 . . . 4 (𝐵 ∈ ℝ* → (0(.g‘ℝ*𝑠)𝐵) = 0)
20 xmul02 11970 . . . 4 (𝐵 ∈ ℝ* → (0 ·e 𝐵) = 0)
2119, 20eqtr4d 2647 . . 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))
2322oveq1d 6564 . . . . 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 786 . . . . . . . 8 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ ℝ*)
26 xrsadd 19582 . . . . . . . . 9 +𝑒 = (+g‘ℝ*𝑠)
2716, 18, 26mulgnnp1 17372 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
2824, 25, 27syl2anc 691 . . . . . . 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 786 . . . . . . . 8 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ 𝑚 = 0) → 𝐵 ∈ ℝ*)
31 xaddid2 11947 . . . . . . . . . 10 (𝐵 ∈ ℝ* → (0 +𝑒 𝐵) = 𝐵)
3231adantl 481 . . . . . . . . 9 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (0 +𝑒 𝐵) = 𝐵)
33 simpl 472 . . . . . . . . . . . 12 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → 𝑚 = 0)
3433oveq1d 6564 . . . . . . . . . . 11 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) = (0(.g‘ℝ*𝑠)𝐵))
3519adantl 481 . . . . . . . . . . 11 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (0(.g‘ℝ*𝑠)𝐵) = 0)
3634, 35eqtrd 2644 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) = 0)
3736oveq1d 6564 . . . . . . . . 9 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵) = (0 +𝑒 𝐵))
3833oveq1d 6564 . . . . . . . . . . . 12 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚 + 1) = (0 + 1))
39 0p1e1 11009 . . . . . . . . . . . 12 (0 + 1) = 1
4038, 39syl6eq 2660 . . . . . . . . . . 11 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (𝑚 + 1) = 1)
4140oveq1d 6564 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = (1(.g‘ℝ*𝑠)𝐵))
4216, 18mulg1 17371 . . . . . . . . . . 11 (𝐵 ∈ ℝ* → (1(.g‘ℝ*𝑠)𝐵) = 𝐵)
4342adantl 481 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → (1(.g‘ℝ*𝑠)𝐵) = 𝐵)
4441, 43eqtrd 2644 . . . . . . . . 9 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = 𝐵)
4532, 37, 443eqtr4rd 2655 . . . . . . . 8 ((𝑚 = 0 ∧ 𝐵 ∈ ℝ*) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
4629, 30, 45syl2anc 691 . . . . . . 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 11171 . . . . . . . 8 (𝑚 ∈ ℕ0 ↔ (𝑚 ∈ ℕ ∨ 𝑚 = 0))
4947, 48sylib 207 . . . . . . 7 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) → (𝑚 ∈ ℕ ∨ 𝑚 = 0))
5028, 46, 49mpjaodan 823 . . . . . 6 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚(.g‘ℝ*𝑠)𝐵) +𝑒 𝐵))
5150adantr 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 11173 . . . . . . . . 9 0 ⊆ ℝ
53 ressxr 9962 . . . . . . . . 9 ℝ ⊆ ℝ*
5452, 53sstri 3577 . . . . . . . 8 0 ⊆ ℝ*
5547adantr 480 . . . . . . . 8 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℕ0)
5654, 55sseldi 3566 . . . . . . 7 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℝ*)
57 nn0ge0 11195 . . . . . . . 8 (𝑚 ∈ ℕ0 → 0 ≤ 𝑚)
5857ad2antlr 759 . . . . . . 7 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 0 ≤ 𝑚)
59 1re 9918 . . . . . . . . 9 1 ∈ ℝ
6059rexri 9976 . . . . . . . 8 1 ∈ ℝ*
6160a1i 11 . . . . . . 7 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 1 ∈ ℝ*)
62 0le1 10430 . . . . . . . 8 0 ≤ 1
6362a1i 11 . . . . . . 7 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 0 ≤ 1)
64 simpll 786 . . . . . . 7 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝐵 ∈ ℝ*)
65 xadddi2r 12000 . . . . . . 7 (((𝑚 ∈ ℝ* ∧ 0 ≤ 𝑚) ∧ (1 ∈ ℝ* ∧ 0 ≤ 1) ∧ 𝐵 ∈ ℝ*) → ((𝑚 +𝑒 1) ·e 𝐵) = ((𝑚 ·e 𝐵) +𝑒 (1 ·e 𝐵)))
6656, 58, 61, 63, 64, 65syl221anc 1329 . . . . . 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)))
6752, 55sseldi 3566 . . . . . . . 8 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 𝑚 ∈ ℝ)
6859a1i 11 . . . . . . . 8 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → 1 ∈ ℝ)
69 rexadd 11937 . . . . . . . 8 ((𝑚 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑚 +𝑒 1) = (𝑚 + 1))
7067, 68, 69syl2anc 691 . . . . . . 7 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (𝑚 +𝑒 1) = (𝑚 + 1))
7170oveq1d 6564 . . . . . 6 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 +𝑒 1) ·e 𝐵) = ((𝑚 + 1) ·e 𝐵))
72 xmulid2 11982 . . . . . . . 8 (𝐵 ∈ ℝ* → (1 ·e 𝐵) = 𝐵)
7364, 72syl 17 . . . . . . 7 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (1 ·e 𝐵) = 𝐵)
7473oveq2d 6565 . . . . . 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))
7566, 71, 743eqtr3d 2652 . . . . 5 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 + 1) ·e 𝐵) = ((𝑚 ·e 𝐵) +𝑒 𝐵))
7623, 51, 753eqtr4d 2654 . . . 4 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ0) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → ((𝑚 + 1)(.g‘ℝ*𝑠)𝐵) = ((𝑚 + 1) ·e 𝐵))
7776exp31 628 . . 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 11912 . . . . . 6 ((𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵) → -𝑒(𝑚(.g‘ℝ*𝑠)𝐵) = -𝑒(𝑚 ·e 𝐵))
7978adantl 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 2610 . . . . . . . . 9 (invg‘ℝ*𝑠) = (invg‘ℝ*𝑠)
8116, 18, 80mulgnegnn 17374 . . . . . . . 8 ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (-𝑚(.g‘ℝ*𝑠)𝐵) = ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)))
8281ancoms 468 . . . . . . 7 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → (-𝑚(.g‘ℝ*𝑠)𝐵) = ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)))
83 xrsex 19580 . . . . . . . . . . . 12 *𝑠 ∈ V
8483a1i 11 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ℝ*𝑠 ∈ V)
85 ssid 3587 . . . . . . . . . . . 12 * ⊆ ℝ*
8685a1i 11 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ℝ* ⊆ ℝ*)
87 simp2 1055 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → 𝑥 ∈ ℝ*)
88 simp3 1056 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → 𝑦 ∈ ℝ*)
8987, 88xaddcld 12003 . . . . . . . . . . 11 ((𝑚 ∈ ℕ ∧ 𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥 +𝑒 𝑦) ∈ ℝ*)
9016, 18, 26, 84, 86, 89mulgnnsubcl 17376 . . . . . . . . . 10 ((𝑚 ∈ ℕ ∧ 𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
91903anidm12 1375 . . . . . . . . 9 ((𝑚 ∈ ℕ ∧ 𝐵 ∈ ℝ*) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
9291ancoms 468 . . . . . . . 8 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → (𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ*)
93 xrsinvgval 29008 . . . . . . . 8 ((𝑚(.g‘ℝ*𝑠)𝐵) ∈ ℝ* → ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
9492, 93syl 17 . . . . . . 7 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → ((invg‘ℝ*𝑠)‘(𝑚(.g‘ℝ*𝑠)𝐵)) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
9582, 94eqtrd 2644 . . . . . 6 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → (-𝑚(.g‘ℝ*𝑠)𝐵) = -𝑒(𝑚(.g‘ℝ*𝑠)𝐵))
9695adantr 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 10904 . . . . . . . . . 10 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
9897adantl 481 . . . . . . . . 9 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
99 rexneg 11916 . . . . . . . . 9 (𝑚 ∈ ℝ → -𝑒𝑚 = -𝑚)
10098, 99syl 17 . . . . . . . 8 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → -𝑒𝑚 = -𝑚)
101100oveq1d 6564 . . . . . . 7 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → (-𝑒𝑚 ·e 𝐵) = (-𝑚 ·e 𝐵))
102 nnssre 10901 . . . . . . . . . 10 ℕ ⊆ ℝ
103102, 53sstri 3577 . . . . . . . . 9 ℕ ⊆ ℝ*
104 simpr 476 . . . . . . . . 9 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
105103, 104sseldi 3566 . . . . . . . 8 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → 𝑚 ∈ ℝ*)
106 simpl 472 . . . . . . . 8 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → 𝐵 ∈ ℝ*)
107 xmulneg1 11971 . . . . . . . 8 ((𝑚 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
108105, 106, 107syl2anc 691 . . . . . . 7 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → (-𝑒𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
109101, 108eqtr3d 2646 . . . . . 6 ((𝐵 ∈ ℝ*𝑚 ∈ ℕ) → (-𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
110109adantr 480 . . . . 5 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (-𝑚 ·e 𝐵) = -𝑒(𝑚 ·e 𝐵))
11179, 96, 1103eqtr4d 2654 . . . 4 (((𝐵 ∈ ℝ*𝑚 ∈ ℕ) ∧ (𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵)) → (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵))
112111exp31 628 . . 3 (𝐵 ∈ ℝ* → (𝑚 ∈ ℕ → ((𝑚(.g‘ℝ*𝑠)𝐵) = (𝑚 ·e 𝐵) → (-𝑚(.g‘ℝ*𝑠)𝐵) = (-𝑚 ·e 𝐵))))
1133, 6, 9, 12, 15, 21, 77, 112zindd 11354 . 2 (𝐵 ∈ ℝ* → (𝐴 ∈ ℤ → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)))
114113impcom 445 1 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  ℝ*cxr 9952   ≤ cle 9954  -cneg 10146  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  -𝑒cxne 11819   +𝑒 cxad 11820   ·e cxmu 11821  ℝ*𝑠cxrs 15983  invgcminusg 17246  .gcmg 17363 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-fz 12198  df-seq 12664  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-tset 15787  df-ple 15788  df-ds 15791  df-0g 15925  df-xrs 15985  df-minusg 17249  df-mulg 17364 This theorem is referenced by:  xrge0mulgnn0  29020  pnfinf  29068
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