Theorem signswrid 31830

Description: The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-Oct-2018.)

Hypotheses

Ref	Expression
signsw.p	⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
signsw.w	⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⨣ ⟩}

Assertion

Ref	Expression
signswrid	⊢ (𝑋 ∈ {-1, 0, 1} → (𝑋 ⨣ 0) = 𝑋)

Distinct variable group:   𝑎,𝑏,𝑋

Allowed substitution hints:   ⨣ (𝑎,𝑏)   𝑊(𝑎,𝑏)

Proof of Theorem signswrid

Step	Hyp	Ref	Expression
1		c0ex 10637	. . . 4 ⊢ 0 ∈ V
2	1	tpid2 4708	. . 3 ⊢ 0 ∈ {-1, 0, 1}
3		signsw.p	. . . 4 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
4	3	signspval 31824	. . 3 ⊢ ((𝑋 ∈ {-1, 0, 1} ∧ 0 ∈ {-1, 0, 1}) → (𝑋 ⨣ 0) = if(0 = 0, 𝑋, 0))
5	2, 4	mpan2 689	. 2 ⊢ (𝑋 ∈ {-1, 0, 1} → (𝑋 ⨣ 0) = if(0 = 0, 𝑋, 0))
6		eqid 2823	. . 3 ⊢ 0 = 0
7		iftrue 4475	. . 3 ⊢ (0 = 0 → if(0 = 0, 𝑋, 0) = 𝑋)
8	6, 7	mp1i 13	. 2 ⊢ (𝑋 ∈ {-1, 0, 1} → if(0 = 0, 𝑋, 0) = 𝑋)
9	5, 8	eqtrd 2858	1 ⊢ (𝑋 ∈ {-1, 0, 1} → (𝑋 ⨣ 0) = 𝑋)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ifcif 4469  {cpr 4571  {ctp 4573  ⟨cop 4575  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  0cc0 10539  1c1 10540  -cneg 10873  ndxcnx 16482  Basecbs 16485  +_gcplusg 16567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-mulcl 10601  ax-i2m1 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163

This theorem is referenced by:  signstfveq0  31849