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Theorem sgnsval 30730
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 30729 . . 3 (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
65adantr 481 . 2 ((𝑅𝑉𝑋𝐵) → 𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
7 eqeq1 2822 . . . 4 (𝑥 = 𝑋 → (𝑥 = 0𝑋 = 0 ))
8 breq2 5061 . . . . 5 (𝑥 = 𝑋 → ( 0 < 𝑥0 < 𝑋))
98ifbid 4485 . . . 4 (𝑥 = 𝑋 → if( 0 < 𝑥, 1, -1) = if( 0 < 𝑋, 1, -1))
107, 9ifbieq2d 4488 . . 3 (𝑥 = 𝑋 → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
1110adantl 482 . 2 (((𝑅𝑉𝑋𝐵) ∧ 𝑥 = 𝑋) → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
12 simpr 485 . 2 ((𝑅𝑉𝑋𝐵) → 𝑋𝐵)
13 c0ex 10623 . . . 4 0 ∈ V
1413a1i 11 . . 3 (((𝑅𝑉𝑋𝐵) ∧ 𝑋 = 0 ) → 0 ∈ V)
15 1ex 10625 . . . . 5 1 ∈ V
1615a1i 11 . . . 4 ((((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) ∧ 0 < 𝑋) → 1 ∈ V)
17 negex 10872 . . . . 5 -1 ∈ V
1817a1i 11 . . . 4 ((((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) ∧ ¬ 0 < 𝑋) → -1 ∈ V)
1916, 18ifclda 4497 . . 3 (((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) → if( 0 < 𝑋, 1, -1) ∈ V)
2014, 19ifclda 4497 . 2 ((𝑅𝑉𝑋𝐵) → if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)) ∈ V)
216, 11, 12, 20fvmptd 6767 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 396   = wceq 1528  wcel 2105  Vcvv 3492  ifcif 4463   class class class wbr 5057  cmpt 5137  cfv 6348  0cc0 10525  1c1 10526  -cneg 10859  Basecbs 16471  0gc0g 16701  ltcplt 17539  sgnscsgns 30727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-neg 10861  df-sgns 30728
This theorem is referenced by: (None)
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