2txmodxeq0 - Metamath Proof Explorer

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Theorem 2txmodxeq0 13117

Description: Two times a positive real number modulo the real number is zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)

**Assertion**
Ref	Expression
2txmodxeq0	⊢ (𝑋 ∈ ℝ⁺ → ((2 · 𝑋) mod 𝑋) = 0)

**Proof of Theorem 2txmodxeq0**
Step	Hyp	Ref	Expression
1		2cnd 11521	. . . 4 ⊢ (𝑋 ∈ ℝ⁺ → 2 ∈ ℂ)
2		rpcn 12219	. . . 4 ⊢ (𝑋 ∈ ℝ⁺ → 𝑋 ∈ ℂ)
3		rpne0 12225	. . . 4 ⊢ (𝑋 ∈ ℝ⁺ → 𝑋 ≠ 0)
4	1, 2, 3	divcan4d 11225	. . 3 ⊢ (𝑋 ∈ ℝ⁺ → ((2 · 𝑋) / 𝑋) = 2)
5		2z 11830	. . 3 ⊢ 2 ∈ ℤ
6	4, 5	syl6eqel 2874	. 2 ⊢ (𝑋 ∈ ℝ⁺ → ((2 · 𝑋) / 𝑋) ∈ ℤ)
7		2re 11517	. . . . 5 ⊢ 2 ∈ ℝ
8	7	a1i 11	. . . 4 ⊢ (𝑋 ∈ ℝ⁺ → 2 ∈ ℝ)
9		rpre 12215	. . . 4 ⊢ (𝑋 ∈ ℝ⁺ → 𝑋 ∈ ℝ)
10	8, 9	remulcld 10472	. . 3 ⊢ (𝑋 ∈ ℝ⁺ → (2 · 𝑋) ∈ ℝ)
11		mod0 13062	. . 3 ⊢ (((2 · 𝑋) ∈ ℝ ∧ 𝑋 ∈ ℝ⁺) → (((2 · 𝑋) mod 𝑋) = 0 ↔ ((2 · 𝑋) / 𝑋) ∈ ℤ))
12	10, 11	mpancom 675	. 2 ⊢ (𝑋 ∈ ℝ⁺ → (((2 · 𝑋) mod 𝑋) = 0 ↔ ((2 · 𝑋) / 𝑋) ∈ ℤ))
13	6, 12	mpbird 249	1 ⊢ (𝑋 ∈ ℝ⁺ → ((2 · 𝑋) mod 𝑋) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 (class class class)co 6978 ℝcr 10336 0cc0 10337 · cmul 10342 / cdiv 11100 2c2 11498 ℤcz 11796 ℝ⁺crp 12207 mod cmo 13055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-sup 8703 df-inf 8704 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-n0 11711 df-z 11797 df-uz 12062 df-rp 12208 df-fl 12980 df-mod 13056

This theorem is referenced by: (None)

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