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Theorem xmulass 12668
Description: Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 12630 which has to avoid the "undefined" combinations +∞ +𝑒 -∞ and -∞ +𝑒 +∞. The equivalent "undefined" expression here would be 0 ·e +∞, but since this is defined to equal 0 any zeroes in the expression make the whole thing evaluate to zero (on both sides), thus establishing the identity in this case. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmulass ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) = (𝐴 ·e (𝐵 ·e 𝐶)))

Proof of Theorem xmulass
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7147 . . . 4 (𝑥 = 𝐴 → (𝑥 ·e 𝐵) = (𝐴 ·e 𝐵))
21oveq1d 7155 . . 3 (𝑥 = 𝐴 → ((𝑥 ·e 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐵) ·e 𝐶))
3 oveq1 7147 . . 3 (𝑥 = 𝐴 → (𝑥 ·e (𝐵 ·e 𝐶)) = (𝐴 ·e (𝐵 ·e 𝐶)))
42, 3eqeq12d 2840 . 2 (𝑥 = 𝐴 → (((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶)) ↔ ((𝐴 ·e 𝐵) ·e 𝐶) = (𝐴 ·e (𝐵 ·e 𝐶))))
5 oveq1 7147 . . . 4 (𝑥 = -𝑒𝐴 → (𝑥 ·e 𝐵) = (-𝑒𝐴 ·e 𝐵))
65oveq1d 7155 . . 3 (𝑥 = -𝑒𝐴 → ((𝑥 ·e 𝐵) ·e 𝐶) = ((-𝑒𝐴 ·e 𝐵) ·e 𝐶))
7 oveq1 7147 . . 3 (𝑥 = -𝑒𝐴 → (𝑥 ·e (𝐵 ·e 𝐶)) = (-𝑒𝐴 ·e (𝐵 ·e 𝐶)))
86, 7eqeq12d 2840 . 2 (𝑥 = -𝑒𝐴 → (((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶)) ↔ ((-𝑒𝐴 ·e 𝐵) ·e 𝐶) = (-𝑒𝐴 ·e (𝐵 ·e 𝐶))))
9 xmulcl 12654 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*)
10 xmulcl 12654 . . 3 (((𝐴 ·e 𝐵) ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
119, 10stoic3 1778 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
12 simp1 1133 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → 𝐴 ∈ ℝ*)
13 xmulcl 12654 . . . 4 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 ·e 𝐶) ∈ ℝ*)
14133adant1 1127 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 ·e 𝐶) ∈ ℝ*)
15 xmulcl 12654 . . 3 ((𝐴 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (𝐴 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
1612, 14, 15syl2anc 587 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐴 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
17 oveq2 7148 . . . . 5 (𝑦 = 𝐵 → (𝑥 ·e 𝑦) = (𝑥 ·e 𝐵))
1817oveq1d 7155 . . . 4 (𝑦 = 𝐵 → ((𝑥 ·e 𝑦) ·e 𝐶) = ((𝑥 ·e 𝐵) ·e 𝐶))
19 oveq1 7147 . . . . 5 (𝑦 = 𝐵 → (𝑦 ·e 𝐶) = (𝐵 ·e 𝐶))
2019oveq2d 7156 . . . 4 (𝑦 = 𝐵 → (𝑥 ·e (𝑦 ·e 𝐶)) = (𝑥 ·e (𝐵 ·e 𝐶)))
2118, 20eqeq12d 2840 . . 3 (𝑦 = 𝐵 → (((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶)) ↔ ((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶))))
22 oveq2 7148 . . . . 5 (𝑦 = -𝑒𝐵 → (𝑥 ·e 𝑦) = (𝑥 ·e -𝑒𝐵))
2322oveq1d 7155 . . . 4 (𝑦 = -𝑒𝐵 → ((𝑥 ·e 𝑦) ·e 𝐶) = ((𝑥 ·e -𝑒𝐵) ·e 𝐶))
24 oveq1 7147 . . . . 5 (𝑦 = -𝑒𝐵 → (𝑦 ·e 𝐶) = (-𝑒𝐵 ·e 𝐶))
2524oveq2d 7156 . . . 4 (𝑦 = -𝑒𝐵 → (𝑥 ·e (𝑦 ·e 𝐶)) = (𝑥 ·e (-𝑒𝐵 ·e 𝐶)))
2623, 25eqeq12d 2840 . . 3 (𝑦 = -𝑒𝐵 → (((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶)) ↔ ((𝑥 ·e -𝑒𝐵) ·e 𝐶) = (𝑥 ·e (-𝑒𝐵 ·e 𝐶))))
27 simprl 770 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝑥 ∈ ℝ*)
28 simpl2 1189 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝐵 ∈ ℝ*)
29 xmulcl 12654 . . . . 5 ((𝑥 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑥 ·e 𝐵) ∈ ℝ*)
3027, 28, 29syl2anc 587 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e 𝐵) ∈ ℝ*)
31 simpl3 1190 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝐶 ∈ ℝ*)
32 xmulcl 12654 . . . 4 (((𝑥 ·e 𝐵) ∈ ℝ*𝐶 ∈ ℝ*) → ((𝑥 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
3330, 31, 32syl2anc 587 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
3414adantr 484 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝐵 ·e 𝐶) ∈ ℝ*)
35 xmulcl 12654 . . . 4 ((𝑥 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
3627, 34, 35syl2anc 587 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
37 oveq2 7148 . . . . 5 (𝑧 = 𝐶 → ((𝑥 ·e 𝑦) ·e 𝑧) = ((𝑥 ·e 𝑦) ·e 𝐶))
38 oveq2 7148 . . . . . 6 (𝑧 = 𝐶 → (𝑦 ·e 𝑧) = (𝑦 ·e 𝐶))
3938oveq2d 7156 . . . . 5 (𝑧 = 𝐶 → (𝑥 ·e (𝑦 ·e 𝑧)) = (𝑥 ·e (𝑦 ·e 𝐶)))
4037, 39eqeq12d 2840 . . . 4 (𝑧 = 𝐶 → (((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)) ↔ ((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶))))
41 oveq2 7148 . . . . 5 (𝑧 = -𝑒𝐶 → ((𝑥 ·e 𝑦) ·e 𝑧) = ((𝑥 ·e 𝑦) ·e -𝑒𝐶))
42 oveq2 7148 . . . . . 6 (𝑧 = -𝑒𝐶 → (𝑦 ·e 𝑧) = (𝑦 ·e -𝑒𝐶))
4342oveq2d 7156 . . . . 5 (𝑧 = -𝑒𝐶 → (𝑥 ·e (𝑦 ·e 𝑧)) = (𝑥 ·e (𝑦 ·e -𝑒𝐶)))
4441, 43eqeq12d 2840 . . . 4 (𝑧 = -𝑒𝐶 → (((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)) ↔ ((𝑥 ·e 𝑦) ·e -𝑒𝐶) = (𝑥 ·e (𝑦 ·e -𝑒𝐶))))
4527adantr 484 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → 𝑥 ∈ ℝ*)
46 simprl 770 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → 𝑦 ∈ ℝ*)
47 xmulcl 12654 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*)
4845, 46, 47syl2anc 587 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e 𝑦) ∈ ℝ*)
4931adantr 484 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → 𝐶 ∈ ℝ*)
50 xmulcl 12654 . . . . 5 (((𝑥 ·e 𝑦) ∈ ℝ*𝐶 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e 𝐶) ∈ ℝ*)
5148, 49, 50syl2anc 587 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 𝐶) ∈ ℝ*)
52 xmulcl 12654 . . . . . 6 ((𝑦 ∈ ℝ*𝐶 ∈ ℝ*) → (𝑦 ·e 𝐶) ∈ ℝ*)
5346, 49, 52syl2anc 587 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑦 ·e 𝐶) ∈ ℝ*)
54 xmulcl 12654 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝑦 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e (𝑦 ·e 𝐶)) ∈ ℝ*)
5545, 53, 54syl2anc 587 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e (𝑦 ·e 𝐶)) ∈ ℝ*)
56 xmulasslem3 12667 . . . . 5 (((𝑥 ∈ ℝ* ∧ 0 < 𝑥) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦) ∧ (𝑧 ∈ ℝ* ∧ 0 < 𝑧)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)))
5756ad4ant234 1172 . . . 4 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) ∧ (𝑧 ∈ ℝ* ∧ 0 < 𝑧)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)))
58 xmul01 12648 . . . . . . . 8 ((𝑥 ·e 𝑦) ∈ ℝ* → ((𝑥 ·e 𝑦) ·e 0) = 0)
5948, 58syl 17 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 0) = 0)
60 xmul01 12648 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑥 ·e 0) = 0)
6145, 60syl 17 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e 0) = 0)
6259, 61eqtr4d 2862 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 0) = (𝑥 ·e 0))
63 xmul01 12648 . . . . . . . 8 (𝑦 ∈ ℝ* → (𝑦 ·e 0) = 0)
6463ad2antrl 727 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑦 ·e 0) = 0)
6564oveq2d 7156 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e (𝑦 ·e 0)) = (𝑥 ·e 0))
6662, 65eqtr4d 2862 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 0) = (𝑥 ·e (𝑦 ·e 0)))
67 oveq2 7148 . . . . . 6 (𝑧 = 0 → ((𝑥 ·e 𝑦) ·e 𝑧) = ((𝑥 ·e 𝑦) ·e 0))
68 oveq2 7148 . . . . . . 7 (𝑧 = 0 → (𝑦 ·e 𝑧) = (𝑦 ·e 0))
6968oveq2d 7156 . . . . . 6 (𝑧 = 0 → (𝑥 ·e (𝑦 ·e 𝑧)) = (𝑥 ·e (𝑦 ·e 0)))
7067, 69eqeq12d 2840 . . . . 5 (𝑧 = 0 → (((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)) ↔ ((𝑥 ·e 𝑦) ·e 0) = (𝑥 ·e (𝑦 ·e 0))))
7166, 70syl5ibrcom 250 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑧 = 0 → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧))))
72 xmulneg2 12651 . . . . 5 (((𝑥 ·e 𝑦) ∈ ℝ*𝐶 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e -𝑒𝐶) = -𝑒((𝑥 ·e 𝑦) ·e 𝐶))
7348, 49, 72syl2anc 587 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e -𝑒𝐶) = -𝑒((𝑥 ·e 𝑦) ·e 𝐶))
74 xmulneg2 12651 . . . . . . 7 ((𝑦 ∈ ℝ*𝐶 ∈ ℝ*) → (𝑦 ·e -𝑒𝐶) = -𝑒(𝑦 ·e 𝐶))
7546, 49, 74syl2anc 587 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑦 ·e -𝑒𝐶) = -𝑒(𝑦 ·e 𝐶))
7675oveq2d 7156 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e (𝑦 ·e -𝑒𝐶)) = (𝑥 ·e -𝑒(𝑦 ·e 𝐶)))
77 xmulneg2 12651 . . . . . 6 ((𝑥 ∈ ℝ* ∧ (𝑦 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e -𝑒(𝑦 ·e 𝐶)) = -𝑒(𝑥 ·e (𝑦 ·e 𝐶)))
7845, 53, 77syl2anc 587 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e -𝑒(𝑦 ·e 𝐶)) = -𝑒(𝑥 ·e (𝑦 ·e 𝐶)))
7976, 78eqtrd 2859 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e (𝑦 ·e -𝑒𝐶)) = -𝑒(𝑥 ·e (𝑦 ·e 𝐶)))
8040, 44, 51, 55, 49, 57, 71, 73, 79xmulasslem 12666 . . 3 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶)))
81 xmul02 12649 . . . . . . . 8 (𝐶 ∈ ℝ* → (0 ·e 𝐶) = 0)
82813ad2ant3 1132 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (0 ·e 𝐶) = 0)
8382adantr 484 . . . . . 6 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (0 ·e 𝐶) = 0)
8460ad2antrl 727 . . . . . 6 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e 0) = 0)
8583, 84eqtr4d 2862 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (0 ·e 𝐶) = (𝑥 ·e 0))
8684oveq1d 7155 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 0) ·e 𝐶) = (0 ·e 𝐶))
8783oveq2d 7156 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (0 ·e 𝐶)) = (𝑥 ·e 0))
8885, 86, 873eqtr4d 2869 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 0) ·e 𝐶) = (𝑥 ·e (0 ·e 𝐶)))
89 oveq2 7148 . . . . . 6 (𝑦 = 0 → (𝑥 ·e 𝑦) = (𝑥 ·e 0))
9089oveq1d 7155 . . . . 5 (𝑦 = 0 → ((𝑥 ·e 𝑦) ·e 𝐶) = ((𝑥 ·e 0) ·e 𝐶))
91 oveq1 7147 . . . . . 6 (𝑦 = 0 → (𝑦 ·e 𝐶) = (0 ·e 𝐶))
9291oveq2d 7156 . . . . 5 (𝑦 = 0 → (𝑥 ·e (𝑦 ·e 𝐶)) = (𝑥 ·e (0 ·e 𝐶)))
9390, 92eqeq12d 2840 . . . 4 (𝑦 = 0 → (((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶)) ↔ ((𝑥 ·e 0) ·e 𝐶) = (𝑥 ·e (0 ·e 𝐶))))
9488, 93syl5ibrcom 250 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑦 = 0 → ((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶))))
95 xmulneg2 12651 . . . . . 6 ((𝑥 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑥 ·e -𝑒𝐵) = -𝑒(𝑥 ·e 𝐵))
9627, 28, 95syl2anc 587 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e -𝑒𝐵) = -𝑒(𝑥 ·e 𝐵))
9796oveq1d 7155 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e -𝑒𝐵) ·e 𝐶) = (-𝑒(𝑥 ·e 𝐵) ·e 𝐶))
98 xmulneg1 12650 . . . . 5 (((𝑥 ·e 𝐵) ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒(𝑥 ·e 𝐵) ·e 𝐶) = -𝑒((𝑥 ·e 𝐵) ·e 𝐶))
9930, 31, 98syl2anc 587 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (-𝑒(𝑥 ·e 𝐵) ·e 𝐶) = -𝑒((𝑥 ·e 𝐵) ·e 𝐶))
10097, 99eqtrd 2859 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e -𝑒𝐵) ·e 𝐶) = -𝑒((𝑥 ·e 𝐵) ·e 𝐶))
101 xmulneg1 12650 . . . . . 6 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒𝐵 ·e 𝐶) = -𝑒(𝐵 ·e 𝐶))
10228, 31, 101syl2anc 587 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (-𝑒𝐵 ·e 𝐶) = -𝑒(𝐵 ·e 𝐶))
103102oveq2d 7156 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (-𝑒𝐵 ·e 𝐶)) = (𝑥 ·e -𝑒(𝐵 ·e 𝐶)))
104 xmulneg2 12651 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e -𝑒(𝐵 ·e 𝐶)) = -𝑒(𝑥 ·e (𝐵 ·e 𝐶)))
10527, 34, 104syl2anc 587 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e -𝑒(𝐵 ·e 𝐶)) = -𝑒(𝑥 ·e (𝐵 ·e 𝐶)))
106103, 105eqtrd 2859 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (-𝑒𝐵 ·e 𝐶)) = -𝑒(𝑥 ·e (𝐵 ·e 𝐶)))
10721, 26, 33, 36, 28, 80, 94, 100, 106xmulasslem 12666 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶)))
108 xmul02 12649 . . . . . 6 (𝐵 ∈ ℝ* → (0 ·e 𝐵) = 0)
1091083ad2ant2 1131 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (0 ·e 𝐵) = 0)
110109oveq1d 7155 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((0 ·e 𝐵) ·e 𝐶) = (0 ·e 𝐶))
111 xmul02 12649 . . . . 5 ((𝐵 ·e 𝐶) ∈ ℝ* → (0 ·e (𝐵 ·e 𝐶)) = 0)
11214, 111syl 17 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (0 ·e (𝐵 ·e 𝐶)) = 0)
11382, 110, 1123eqtr4d 2869 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((0 ·e 𝐵) ·e 𝐶) = (0 ·e (𝐵 ·e 𝐶)))
114 oveq1 7147 . . . . 5 (𝑥 = 0 → (𝑥 ·e 𝐵) = (0 ·e 𝐵))
115114oveq1d 7155 . . . 4 (𝑥 = 0 → ((𝑥 ·e 𝐵) ·e 𝐶) = ((0 ·e 𝐵) ·e 𝐶))
116 oveq1 7147 . . . 4 (𝑥 = 0 → (𝑥 ·e (𝐵 ·e 𝐶)) = (0 ·e (𝐵 ·e 𝐶)))
117115, 116eqeq12d 2840 . . 3 (𝑥 = 0 → (((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶)) ↔ ((0 ·e 𝐵) ·e 𝐶) = (0 ·e (𝐵 ·e 𝐶))))
118113, 117syl5ibrcom 250 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝑥 = 0 → ((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶))))
119 xmulneg1 12650 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
1201193adant3 1129 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
121120oveq1d 7155 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((-𝑒𝐴 ·e 𝐵) ·e 𝐶) = (-𝑒(𝐴 ·e 𝐵) ·e 𝐶))
122 xmulneg1 12650 . . . 4 (((𝐴 ·e 𝐵) ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒(𝐴 ·e 𝐵) ·e 𝐶) = -𝑒((𝐴 ·e 𝐵) ·e 𝐶))
1239, 122stoic3 1778 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒(𝐴 ·e 𝐵) ·e 𝐶) = -𝑒((𝐴 ·e 𝐵) ·e 𝐶))
124121, 123eqtrd 2859 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((-𝑒𝐴 ·e 𝐵) ·e 𝐶) = -𝑒((𝐴 ·e 𝐵) ·e 𝐶))
125 xmulneg1 12650 . . 3 ((𝐴 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (-𝑒𝐴 ·e (𝐵 ·e 𝐶)) = -𝑒(𝐴 ·e (𝐵 ·e 𝐶)))
12612, 14, 125syl2anc 587 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒𝐴 ·e (𝐵 ·e 𝐶)) = -𝑒(𝐴 ·e (𝐵 ·e 𝐶)))
1274, 8, 11, 16, 12, 107, 118, 124, 126xmulasslem 12666 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) = (𝐴 ·e (𝐵 ·e 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115   class class class wbr 5049  (class class class)co 7140  0cc0 10524  *cxr 10661   < clt 10662  -𝑒cxne 12492   ·e cxmu 12494
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-cnex 10580  ax-resscn 10581  ax-1cn 10582  ax-icn 10583  ax-addcl 10584  ax-addrcl 10585  ax-mulcl 10586  ax-mulrcl 10587  ax-mulcom 10588  ax-addass 10589  ax-mulass 10590  ax-distr 10591  ax-i2m1 10592  ax-1ne0 10593  ax-1rid 10594  ax-rnegex 10595  ax-rrecex 10596  ax-cnre 10597  ax-pre-lttri 10598  ax-pre-lttrn 10599  ax-pre-ltadd 10600  ax-pre-mulgt0 10601
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-po 5457  df-so 5458  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7674  df-2nd 7675  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-pnf 10664  df-mnf 10665  df-xr 10666  df-ltxr 10667  df-le 10668  df-sub 10859  df-neg 10860  df-xneg 12495  df-xmul 12497
This theorem is referenced by:  xlemul1  12671  xrsmcmn  20556  nmoi2  23327  xmulcand  30596  xreceu  30597  xdivrec  30602  xrge0slmod  30937
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