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Theorem xadddi2r 13353

Description: Commuted version of xadddi2 13352. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xadddi2r	⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)))

**Proof of Theorem xadddi2r**
Step	Hyp	Ref	Expression
1		xadddi2 13352	. . 3 ⊢ ((𝐶 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵)) → (𝐶 ·_e (𝐴 +_𝑒 𝐵)) = ((𝐶 ·_e 𝐴) +_𝑒 (𝐶 ·_e 𝐵)))
2	1	3coml 1145	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐶 ·_e (𝐴 +_𝑒 𝐵)) = ((𝐶 ·_e 𝐴) +_𝑒 (𝐶 ·_e 𝐵)))
3		simp1l 1216	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		simp2l 1218	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
5		xaddcl 13294	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
6	3, 4, 5	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
7		simp3 1156	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
8		xmulcom 13321	. . 3 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (𝐶 ·_e (𝐴 +_𝑒 𝐵)))
9	6, 7, 8	syl2anc 596	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (𝐶 ·_e (𝐴 +_𝑒 𝐵)))
10		xmulcom 13321	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e 𝐶) = (𝐶 ·_e 𝐴))
11	3, 7, 10	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e 𝐶) = (𝐶 ·_e 𝐴))
12		xmulcom 13321	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ·_e 𝐶) = (𝐶 ·_e 𝐵))
13	4, 7, 12	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐵 ·_e 𝐶) = (𝐶 ·_e 𝐵))
14	11, 13	oveq12d 7432	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)) = ((𝐶 ·_e 𝐴) +_𝑒 (𝐶 ·_e 𝐵)))
15	2, 9, 14	3eqtr4d 2805	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = ((𝐴 ·_e 𝐶) +_𝑒 (𝐵 ·_e 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 class class class wbr 5103 (class class class)co 7414 0cc0 11127 ℝ^*cxr 11269 ≤ cle 11271 +_𝑒 cxad 13164 ·_e cxmu 13165

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-xneg 13166 df-xadd 13167 df-xmul 13168

This theorem is used by: x2times 13354 xrsmulgzz 33452

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