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Theorem xadddi2r 12341

Description: Commuted version of xadddi2 12340. (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 12340	. . 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 1122	. 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 1240	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		simp2l 1242	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
5		xaddcl 12283	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
6	3, 4, 5	syl2anc 696	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
7		simp3 1133	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
8		xmulcom 12309	. . 3 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (𝐶 ·_e (𝐴 +_𝑒 𝐵)))
9	6, 7, 8	syl2anc 696	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (𝐶 ·_e (𝐴 +_𝑒 𝐵)))
10		xmulcom 12309	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e 𝐶) = (𝐶 ·_e 𝐴))
11	3, 7, 10	syl2anc 696	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e 𝐶) = (𝐶 ·_e 𝐴))
12		xmulcom 12309	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ·_e 𝐶) = (𝐶 ·_e 𝐵))
13	4, 7, 12	syl2anc 696	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐵 ·_e 𝐶) = (𝐶 ·_e 𝐵))
14	11, 13	oveq12d 6832	. 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 2804	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

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 (class class class)co 6814 0cc0 10148 ℝ^*cxr 10285 ≤ cle 10287 +_𝑒 cxad 12157 ·_e cxmu 12158

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-1st 7334 df-2nd 7335 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-xneg 12159 df-xadd 12160 df-xmul 12161

This theorem is referenced by: x2times 12342 xrsmulgzz 30008

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