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Theorem sgnsval 33164
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 33163 . . 3 (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
65adantr 480 . 2 ((𝑅𝑉𝑋𝐵) → 𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
7 eqeq1 2739 . . . 4 (𝑥 = 𝑋 → (𝑥 = 0𝑋 = 0 ))
8 breq2 5152 . . . . 5 (𝑥 = 𝑋 → ( 0 < 𝑥0 < 𝑋))
98ifbid 4554 . . . 4 (𝑥 = 𝑋 → if( 0 < 𝑥, 1, -1) = if( 0 < 𝑋, 1, -1))
107, 9ifbieq2d 4557 . . 3 (𝑥 = 𝑋 → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
1110adantl 481 . 2 (((𝑅𝑉𝑋𝐵) ∧ 𝑥 = 𝑋) → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
12 simpr 484 . 2 ((𝑅𝑉𝑋𝐵) → 𝑋𝐵)
13 c0ex 11253 . . . 4 0 ∈ V
1413a1i 11 . . 3 (((𝑅𝑉𝑋𝐵) ∧ 𝑋 = 0 ) → 0 ∈ V)
15 1ex 11255 . . . . 5 1 ∈ V
1615a1i 11 . . . 4 ((((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) ∧ 0 < 𝑋) → 1 ∈ V)
17 negex 11504 . . . . 5 -1 ∈ V
1817a1i 11 . . . 4 ((((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) ∧ ¬ 0 < 𝑋) → -1 ∈ V)
1916, 18ifclda 4566 . . 3 (((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) → if( 0 < 𝑋, 1, -1) ∈ V)
2014, 19ifclda 4566 . 2 ((𝑅𝑉𝑋𝐵) → if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)) ∈ V)
216, 11, 12, 20fvmptd 7023 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 395   = wceq 1537  wcel 2106  Vcvv 3478  ifcif 4531   class class class wbr 5148  cmpt 5231  cfv 6563  0cc0 11153  1c1 11154  -cneg 11491  Basecbs 17245  0gc0g 17486  ltcplt 18366  sgnscsgns 33161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-mulcl 11215  ax-i2m1 11221
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-neg 11493  df-sgns 33162
This theorem is referenced by: (None)
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