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Theorem sgnsval 30058
Description: The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.)
Hypotheses
Ref Expression
sgnsval.b 𝐵 = (Base‘𝑅)
sgnsval.0 0 = (0g𝑅)
sgnsval.l < = (lt‘𝑅)
sgnsval.s 𝑆 = (sgns𝑅)
Assertion
Ref Expression
sgnsval ((𝑅𝑉𝑋𝐵) → (𝑆𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))

Proof of Theorem sgnsval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sgnsval.b . . . 4 𝐵 = (Base‘𝑅)
2 sgnsval.0 . . . 4 0 = (0g𝑅)
3 sgnsval.l . . . 4 < = (lt‘𝑅)
4 sgnsval.s . . . 4 𝑆 = (sgns𝑅)
51, 2, 3, 4sgnsv 30057 . . 3 (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
65adantr 472 . 2 ((𝑅𝑉𝑋𝐵) → 𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
7 eqeq1 2764 . . . 4 (𝑥 = 𝑋 → (𝑥 = 0𝑋 = 0 ))
8 breq2 4808 . . . . 5 (𝑥 = 𝑋 → ( 0 < 𝑥0 < 𝑋))
98ifbid 4252 . . . 4 (𝑥 = 𝑋 → if( 0 < 𝑥, 1, -1) = if( 0 < 𝑋, 1, -1))
107, 9ifbieq2d 4255 . . 3 (𝑥 = 𝑋 → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
1110adantl 473 . 2 (((𝑅𝑉𝑋𝐵) ∧ 𝑥 = 𝑋) → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
12 simpr 479 . 2 ((𝑅𝑉𝑋𝐵) → 𝑋𝐵)
13 c0ex 10246 . . . 4 0 ∈ V
1413a1i 11 . . 3 (((𝑅𝑉𝑋𝐵) ∧ 𝑋 = 0 ) → 0 ∈ V)
15 1ex 10247 . . . . 5 1 ∈ V
1615a1i 11 . . . 4 ((((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) ∧ 0 < 𝑋) → 1 ∈ V)
17 negex 10491 . . . . 5 -1 ∈ V
1817a1i 11 . . . 4 ((((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) ∧ ¬ 0 < 𝑋) → -1 ∈ V)
1916, 18ifclda 4264 . . 3 (((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) → if( 0 < 𝑋, 1, -1) ∈ V)
2014, 19ifclda 4264 . 2 ((𝑅𝑉𝑋𝐵) → if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)) ∈ V)
216, 11, 12, 20fvmptd 6451 1 ((𝑅𝑉𝑋𝐵) → (𝑆𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  ifcif 4230   class class class wbr 4804  cmpt 4881  cfv 6049  0cc0 10148  1c1 10149  -cneg 10479  Basecbs 16079  0gc0g 16322  ltcplt 17162  sgnscsgns 30055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-mulcl 10210  ax-i2m1 10216
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-neg 10481  df-sgns 30056
This theorem is referenced by: (None)
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