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Theorem xmulass 12323
Description: Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 12285 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 6801 . . . 4 (𝑥 = 𝐴 → (𝑥 ·e 𝐵) = (𝐴 ·e 𝐵))
21oveq1d 6809 . . 3 (𝑥 = 𝐴 → ((𝑥 ·e 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐵) ·e 𝐶))
3 oveq1 6801 . . 3 (𝑥 = 𝐴 → (𝑥 ·e (𝐵 ·e 𝐶)) = (𝐴 ·e (𝐵 ·e 𝐶)))
42, 3eqeq12d 2786 . 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 6801 . . . 4 (𝑥 = -𝑒𝐴 → (𝑥 ·e 𝐵) = (-𝑒𝐴 ·e 𝐵))
65oveq1d 6809 . . 3 (𝑥 = -𝑒𝐴 → ((𝑥 ·e 𝐵) ·e 𝐶) = ((-𝑒𝐴 ·e 𝐵) ·e 𝐶))
7 oveq1 6801 . . 3 (𝑥 = -𝑒𝐴 → (𝑥 ·e (𝐵 ·e 𝐶)) = (-𝑒𝐴 ·e (𝐵 ·e 𝐶)))
86, 7eqeq12d 2786 . 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 12309 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*)
10 xmulcl 12309 . . 3 (((𝐴 ·e 𝐵) ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
119, 10stoic3 1849 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
12 simp1 1130 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → 𝐴 ∈ ℝ*)
13 xmulcl 12309 . . . 4 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 ·e 𝐶) ∈ ℝ*)
14133adant1 1124 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐵 ·e 𝐶) ∈ ℝ*)
15 xmulcl 12309 . . 3 ((𝐴 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (𝐴 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
1612, 14, 15syl2anc 567 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐴 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
17 oveq2 6802 . . . . 5 (𝑦 = 𝐵 → (𝑥 ·e 𝑦) = (𝑥 ·e 𝐵))
1817oveq1d 6809 . . . 4 (𝑦 = 𝐵 → ((𝑥 ·e 𝑦) ·e 𝐶) = ((𝑥 ·e 𝐵) ·e 𝐶))
19 oveq1 6801 . . . . 5 (𝑦 = 𝐵 → (𝑦 ·e 𝐶) = (𝐵 ·e 𝐶))
2019oveq2d 6810 . . . 4 (𝑦 = 𝐵 → (𝑥 ·e (𝑦 ·e 𝐶)) = (𝑥 ·e (𝐵 ·e 𝐶)))
2118, 20eqeq12d 2786 . . 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 6802 . . . . 5 (𝑦 = -𝑒𝐵 → (𝑥 ·e 𝑦) = (𝑥 ·e -𝑒𝐵))
2322oveq1d 6809 . . . 4 (𝑦 = -𝑒𝐵 → ((𝑥 ·e 𝑦) ·e 𝐶) = ((𝑥 ·e -𝑒𝐵) ·e 𝐶))
24 oveq1 6801 . . . . 5 (𝑦 = -𝑒𝐵 → (𝑦 ·e 𝐶) = (-𝑒𝐵 ·e 𝐶))
2524oveq2d 6810 . . . 4 (𝑦 = -𝑒𝐵 → (𝑥 ·e (𝑦 ·e 𝐶)) = (𝑥 ·e (-𝑒𝐵 ·e 𝐶)))
2623, 25eqeq12d 2786 . . 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 748 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝑥 ∈ ℝ*)
28 simpl2 1229 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝐵 ∈ ℝ*)
29 xmulcl 12309 . . . . 5 ((𝑥 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑥 ·e 𝐵) ∈ ℝ*)
3027, 28, 29syl2anc 567 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e 𝐵) ∈ ℝ*)
31 simpl3 1231 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝐶 ∈ ℝ*)
32 xmulcl 12309 . . . 4 (((𝑥 ·e 𝐵) ∈ ℝ*𝐶 ∈ ℝ*) → ((𝑥 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
3330, 31, 32syl2anc 567 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 𝐵) ·e 𝐶) ∈ ℝ*)
3414adantr 466 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝐵 ·e 𝐶) ∈ ℝ*)
35 xmulcl 12309 . . . 4 ((𝑥 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
3627, 34, 35syl2anc 567 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (𝐵 ·e 𝐶)) ∈ ℝ*)
37 oveq2 6802 . . . . 5 (𝑧 = 𝐶 → ((𝑥 ·e 𝑦) ·e 𝑧) = ((𝑥 ·e 𝑦) ·e 𝐶))
38 oveq2 6802 . . . . . 6 (𝑧 = 𝐶 → (𝑦 ·e 𝑧) = (𝑦 ·e 𝐶))
3938oveq2d 6810 . . . . 5 (𝑧 = 𝐶 → (𝑥 ·e (𝑦 ·e 𝑧)) = (𝑥 ·e (𝑦 ·e 𝐶)))
4037, 39eqeq12d 2786 . . . 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 6802 . . . . 5 (𝑧 = -𝑒𝐶 → ((𝑥 ·e 𝑦) ·e 𝑧) = ((𝑥 ·e 𝑦) ·e -𝑒𝐶))
42 oveq2 6802 . . . . . 6 (𝑧 = -𝑒𝐶 → (𝑦 ·e 𝑧) = (𝑦 ·e -𝑒𝐶))
4342oveq2d 6810 . . . . 5 (𝑧 = -𝑒𝐶 → (𝑥 ·e (𝑦 ·e 𝑧)) = (𝑥 ·e (𝑦 ·e -𝑒𝐶)))
4441, 43eqeq12d 2786 . . . 4 (𝑧 = -𝑒𝐶 → (((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)) ↔ ((𝑥 ·e 𝑦) ·e -𝑒𝐶) = (𝑥 ·e (𝑦 ·e -𝑒𝐶))))
4527adantr 466 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → 𝑥 ∈ ℝ*)
46 simprl 748 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → 𝑦 ∈ ℝ*)
47 xmulcl 12309 . . . . . 6 ((𝑥 ∈ ℝ*𝑦 ∈ ℝ*) → (𝑥 ·e 𝑦) ∈ ℝ*)
4845, 46, 47syl2anc 567 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e 𝑦) ∈ ℝ*)
4931adantr 466 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → 𝐶 ∈ ℝ*)
50 xmulcl 12309 . . . . 5 (((𝑥 ·e 𝑦) ∈ ℝ*𝐶 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e 𝐶) ∈ ℝ*)
5148, 49, 50syl2anc 567 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 𝐶) ∈ ℝ*)
52 xmulcl 12309 . . . . . 6 ((𝑦 ∈ ℝ*𝐶 ∈ ℝ*) → (𝑦 ·e 𝐶) ∈ ℝ*)
5346, 49, 52syl2anc 567 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑦 ·e 𝐶) ∈ ℝ*)
54 xmulcl 12309 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝑦 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e (𝑦 ·e 𝐶)) ∈ ℝ*)
5545, 53, 54syl2anc 567 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e (𝑦 ·e 𝐶)) ∈ ℝ*)
56 xmulasslem3 12322 . . . . . 6 (((𝑥 ∈ ℝ* ∧ 0 < 𝑥) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦) ∧ (𝑧 ∈ ℝ* ∧ 0 < 𝑧)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)))
57563expa 1111 . . . . 5 ((((𝑥 ∈ ℝ* ∧ 0 < 𝑥) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) ∧ (𝑧 ∈ ℝ* ∧ 0 < 𝑧)) → ((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)))
5857adantlll 691 . . . 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)))
59 xmul01 12303 . . . . . . . 8 ((𝑥 ·e 𝑦) ∈ ℝ* → ((𝑥 ·e 𝑦) ·e 0) = 0)
6048, 59syl 17 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 0) = 0)
61 xmul01 12303 . . . . . . . 8 (𝑥 ∈ ℝ* → (𝑥 ·e 0) = 0)
6245, 61syl 17 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e 0) = 0)
6360, 62eqtr4d 2808 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → ((𝑥 ·e 𝑦) ·e 0) = (𝑥 ·e 0))
64 xmul01 12303 . . . . . . . 8 (𝑦 ∈ ℝ* → (𝑦 ·e 0) = 0)
6564ad2antrl 701 . . . . . . 7 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑦 ·e 0) = 0)
6665oveq2d 6810 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑥 ·e (𝑦 ·e 0)) = (𝑥 ·e 0))
6763, 66eqtr4d 2808 . . . . 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)))
68 oveq2 6802 . . . . . 6 (𝑧 = 0 → ((𝑥 ·e 𝑦) ·e 𝑧) = ((𝑥 ·e 𝑦) ·e 0))
69 oveq2 6802 . . . . . . 7 (𝑧 = 0 → (𝑦 ·e 𝑧) = (𝑦 ·e 0))
7069oveq2d 6810 . . . . . 6 (𝑧 = 0 → (𝑥 ·e (𝑦 ·e 𝑧)) = (𝑥 ·e (𝑦 ·e 0)))
7168, 70eqeq12d 2786 . . . . 5 (𝑧 = 0 → (((𝑥 ·e 𝑦) ·e 𝑧) = (𝑥 ·e (𝑦 ·e 𝑧)) ↔ ((𝑥 ·e 𝑦) ·e 0) = (𝑥 ·e (𝑦 ·e 0))))
7267, 71syl5ibrcom 237 . . . 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))))
73 xmulneg2 12306 . . . . 5 (((𝑥 ·e 𝑦) ∈ ℝ*𝐶 ∈ ℝ*) → ((𝑥 ·e 𝑦) ·e -𝑒𝐶) = -𝑒((𝑥 ·e 𝑦) ·e 𝐶))
7448, 49, 73syl2anc 567 . . . 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))
75 xmulneg2 12306 . . . . . . 7 ((𝑦 ∈ ℝ*𝐶 ∈ ℝ*) → (𝑦 ·e -𝑒𝐶) = -𝑒(𝑦 ·e 𝐶))
7646, 49, 75syl2anc 567 . . . . . 6 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) ∧ (𝑦 ∈ ℝ* ∧ 0 < 𝑦)) → (𝑦 ·e -𝑒𝐶) = -𝑒(𝑦 ·e 𝐶))
7776oveq2d 6810 . . . . 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)))
78 xmulneg2 12306 . . . . . 6 ((𝑥 ∈ ℝ* ∧ (𝑦 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e -𝑒(𝑦 ·e 𝐶)) = -𝑒(𝑥 ·e (𝑦 ·e 𝐶)))
7945, 53, 78syl2anc 567 . . . . 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)))
8077, 79eqtrd 2805 . . . 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)))
8140, 44, 51, 55, 49, 58, 72, 74, 80xmulasslem 12321 . . 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)))
82 xmul02 12304 . . . . . . . 8 (𝐶 ∈ ℝ* → (0 ·e 𝐶) = 0)
83823ad2ant3 1129 . . . . . . 7 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (0 ·e 𝐶) = 0)
8483adantr 466 . . . . . 6 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (0 ·e 𝐶) = 0)
8561ad2antrl 701 . . . . . 6 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e 0) = 0)
8684, 85eqtr4d 2808 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (0 ·e 𝐶) = (𝑥 ·e 0))
8785oveq1d 6809 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 0) ·e 𝐶) = (0 ·e 𝐶))
8884oveq2d 6810 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (0 ·e 𝐶)) = (𝑥 ·e 0))
8986, 87, 883eqtr4d 2815 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 0) ·e 𝐶) = (𝑥 ·e (0 ·e 𝐶)))
90 oveq2 6802 . . . . . 6 (𝑦 = 0 → (𝑥 ·e 𝑦) = (𝑥 ·e 0))
9190oveq1d 6809 . . . . 5 (𝑦 = 0 → ((𝑥 ·e 𝑦) ·e 𝐶) = ((𝑥 ·e 0) ·e 𝐶))
92 oveq1 6801 . . . . . 6 (𝑦 = 0 → (𝑦 ·e 𝐶) = (0 ·e 𝐶))
9392oveq2d 6810 . . . . 5 (𝑦 = 0 → (𝑥 ·e (𝑦 ·e 𝐶)) = (𝑥 ·e (0 ·e 𝐶)))
9491, 93eqeq12d 2786 . . . 4 (𝑦 = 0 → (((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶)) ↔ ((𝑥 ·e 0) ·e 𝐶) = (𝑥 ·e (0 ·e 𝐶))))
9589, 94syl5ibrcom 237 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑦 = 0 → ((𝑥 ·e 𝑦) ·e 𝐶) = (𝑥 ·e (𝑦 ·e 𝐶))))
96 xmulneg2 12306 . . . . . 6 ((𝑥 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑥 ·e -𝑒𝐵) = -𝑒(𝑥 ·e 𝐵))
9727, 28, 96syl2anc 567 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e -𝑒𝐵) = -𝑒(𝑥 ·e 𝐵))
9897oveq1d 6809 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e -𝑒𝐵) ·e 𝐶) = (-𝑒(𝑥 ·e 𝐵) ·e 𝐶))
99 xmulneg1 12305 . . . . 5 (((𝑥 ·e 𝐵) ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒(𝑥 ·e 𝐵) ·e 𝐶) = -𝑒((𝑥 ·e 𝐵) ·e 𝐶))
10030, 31, 99syl2anc 567 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (-𝑒(𝑥 ·e 𝐵) ·e 𝐶) = -𝑒((𝑥 ·e 𝐵) ·e 𝐶))
10198, 100eqtrd 2805 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e -𝑒𝐵) ·e 𝐶) = -𝑒((𝑥 ·e 𝐵) ·e 𝐶))
102 xmulneg1 12305 . . . . . 6 ((𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒𝐵 ·e 𝐶) = -𝑒(𝐵 ·e 𝐶))
10328, 31, 102syl2anc 567 . . . . 5 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (-𝑒𝐵 ·e 𝐶) = -𝑒(𝐵 ·e 𝐶))
104103oveq2d 6810 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (-𝑒𝐵 ·e 𝐶)) = (𝑥 ·e -𝑒(𝐵 ·e 𝐶)))
105 xmulneg2 12306 . . . . 5 ((𝑥 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (𝑥 ·e -𝑒(𝐵 ·e 𝐶)) = -𝑒(𝑥 ·e (𝐵 ·e 𝐶)))
10627, 34, 105syl2anc 567 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e -𝑒(𝐵 ·e 𝐶)) = -𝑒(𝑥 ·e (𝐵 ·e 𝐶)))
107104, 106eqtrd 2805 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → (𝑥 ·e (-𝑒𝐵 ·e 𝐶)) = -𝑒(𝑥 ·e (𝐵 ·e 𝐶)))
10821, 26, 33, 36, 28, 81, 95, 101, 107xmulasslem 12321 . 2 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → ((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶)))
109 xmul02 12304 . . . . . 6 (𝐵 ∈ ℝ* → (0 ·e 𝐵) = 0)
1101093ad2ant2 1128 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (0 ·e 𝐵) = 0)
111110oveq1d 6809 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((0 ·e 𝐵) ·e 𝐶) = (0 ·e 𝐶))
112 xmul02 12304 . . . . 5 ((𝐵 ·e 𝐶) ∈ ℝ* → (0 ·e (𝐵 ·e 𝐶)) = 0)
11314, 112syl 17 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (0 ·e (𝐵 ·e 𝐶)) = 0)
11483, 111, 1133eqtr4d 2815 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((0 ·e 𝐵) ·e 𝐶) = (0 ·e (𝐵 ·e 𝐶)))
115 oveq1 6801 . . . . 5 (𝑥 = 0 → (𝑥 ·e 𝐵) = (0 ·e 𝐵))
116115oveq1d 6809 . . . 4 (𝑥 = 0 → ((𝑥 ·e 𝐵) ·e 𝐶) = ((0 ·e 𝐵) ·e 𝐶))
117 oveq1 6801 . . . 4 (𝑥 = 0 → (𝑥 ·e (𝐵 ·e 𝐶)) = (0 ·e (𝐵 ·e 𝐶)))
118116, 117eqeq12d 2786 . . 3 (𝑥 = 0 → (((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶)) ↔ ((0 ·e 𝐵) ·e 𝐶) = (0 ·e (𝐵 ·e 𝐶))))
119114, 118syl5ibrcom 237 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝑥 = 0 → ((𝑥 ·e 𝐵) ·e 𝐶) = (𝑥 ·e (𝐵 ·e 𝐶))))
120 xmulneg1 12305 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
1211203adant3 1126 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒𝐴 ·e 𝐵) = -𝑒(𝐴 ·e 𝐵))
122121oveq1d 6809 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((-𝑒𝐴 ·e 𝐵) ·e 𝐶) = (-𝑒(𝐴 ·e 𝐵) ·e 𝐶))
123 xmulneg1 12305 . . . 4 (((𝐴 ·e 𝐵) ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒(𝐴 ·e 𝐵) ·e 𝐶) = -𝑒((𝐴 ·e 𝐵) ·e 𝐶))
1249, 123stoic3 1849 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒(𝐴 ·e 𝐵) ·e 𝐶) = -𝑒((𝐴 ·e 𝐵) ·e 𝐶))
125122, 124eqtrd 2805 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((-𝑒𝐴 ·e 𝐵) ·e 𝐶) = -𝑒((𝐴 ·e 𝐵) ·e 𝐶))
126 xmulneg1 12305 . . 3 ((𝐴 ∈ ℝ* ∧ (𝐵 ·e 𝐶) ∈ ℝ*) → (-𝑒𝐴 ·e (𝐵 ·e 𝐶)) = -𝑒(𝐴 ·e (𝐵 ·e 𝐶)))
12712, 14, 126syl2anc 567 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (-𝑒𝐴 ·e (𝐵 ·e 𝐶)) = -𝑒(𝐴 ·e (𝐵 ·e 𝐶)))
1284, 8, 11, 16, 12, 108, 119, 125, 127xmulasslem 12321 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 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4787  (class class class)co 6794  0cc0 10139  *cxr 10276   < clt 10277  -𝑒cxne 12149   ·e cxmu 12151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-po 5171  df-so 5172  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-1st 7316  df-2nd 7317  df-er 7897  df-en 8111  df-dom 8112  df-sdom 8113  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-xneg 12152  df-xmul 12154
This theorem is referenced by:  xlemul1  12326  xrsmcmn  19985  nmoi2  22755  xmulcand  29970  xreceu  29971  xdivrec  29976  xrge0slmod  30185
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