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Theorem xadddi2r 13250

Description: Commuted version of xadddi2 13249. (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 13249	. . 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 1128	. 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 1199	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		simp2l 1201	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐵 ∈ ℝ^)
5		xaddcl 13191	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
6	3, 4, 5	syl2anc 585	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
7		simp3 1139	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^) → 𝐶 ∈ ℝ^)
8		xmulcom 13218	. . 3 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (𝐶 ·_e (𝐴 +_𝑒 𝐵)))
9	6, 7, 8	syl2anc 585	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) ·_e 𝐶) = (𝐶 ·_e (𝐴 +_𝑒 𝐵)))
10		xmulcom 13218	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e 𝐶) = (𝐶 ·_e 𝐴))
11	3, 7, 10	syl2anc 585	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐴 ·_e 𝐶) = (𝐶 ·_e 𝐴))
12		xmulcom 13218	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ·_e 𝐶) = (𝐶 ·_e 𝐵))
13	4, 7, 12	syl2anc 585	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ^*) → (𝐵 ·_e 𝐶) = (𝐶 ·_e 𝐵))
14	11, 13	oveq12d 7385	. 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 2781	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 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5085 (class class class)co 7367 0cc0 11038 ℝ^*cxr 11178 ≤ cle 11180 +_𝑒 cxad 13061 ·_e cxmu 13062

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-xneg 13063 df-xadd 13064 df-xmul 13065

This theorem is referenced by: x2times 13251 xrsmulgzz 33069

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