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Theorem xadddi2r 13265

Description: Commuted version of xadddi2 13264. (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 13264	. . 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 1127	. 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 1198	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		simp2l 1200	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
5		xaddcl 13206	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
6	3, 4, 5	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
7		simp3 1138	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
8		xmulcom 13233	. . 3 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (𝐶 ·_e (𝐴 +_𝑒 𝐵)))
9	6, 7, 8	syl2anc 584	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (𝐶 ·_e (𝐴 +_𝑒 𝐵)))
10		xmulcom 13233	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e 𝐶) = (𝐶 ·_e 𝐴))
11	3, 7, 10	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e 𝐶) = (𝐶 ·_e 𝐴))
12		xmulcom 13233	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ·_e 𝐶) = (𝐶 ·_e 𝐵))
13	4, 7, 12	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐵 ·_e 𝐶) = (𝐶 ·_e 𝐵))
14	11, 13	oveq12d 7408	. 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 2775	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 class class class wbr 5110 (class class class)co 7390 0cc0 11075 ℝ^*cxr 11214 ≤ cle 11216 +_𝑒 cxad 13077 ·_e cxmu 13078

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-xneg 13079 df-xadd 13080 df-xmul 13081

This theorem is referenced by: x2times 13266 xrsmulgzz 32954

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