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Theorem xadddi2r 12685

Description: Commuted version of xadddi2 12684. (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 12684	. . 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 1123	. 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 1193	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		simp2l 1195	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
5		xaddcl 12626	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
6	3, 4, 5	syl2anc 586	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
7		simp3 1134	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
8		xmulcom 12653	. . 3 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (𝐶 ·_e (𝐴 +_𝑒 𝐵)))
9	6, 7, 8	syl2anc 586	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (𝐶 ·_e (𝐴 +_𝑒 𝐵)))
10		xmulcom 12653	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e 𝐶) = (𝐶 ·_e 𝐴))
11	3, 7, 10	syl2anc 586	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e 𝐶) = (𝐶 ·_e 𝐴))
12		xmulcom 12653	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ·_e 𝐶) = (𝐶 ·_e 𝐵))
13	4, 7, 12	syl2anc 586	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐵 ·_e 𝐶) = (𝐶 ·_e 𝐵))
14	11, 13	oveq12d 7168	. 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 2866	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 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 (class class class)co 7150 0cc0 10531 ℝ^*cxr 10668 ≤ cle 10670 +_𝑒 cxad 12499 ·_e cxmu 12500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-xneg 12501 df-xadd 12502 df-xmul 12503

This theorem is referenced by: x2times 12686 xrsmulgzz 30660

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