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Theorem sgnsval 32974
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 32973 . . 3 (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
65adantr 479 . 2 ((𝑅𝑉𝑋𝐵) → 𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
7 eqeq1 2729 . . . 4 (𝑥 = 𝑋 → (𝑥 = 0𝑋 = 0 ))
8 breq2 5153 . . . . 5 (𝑥 = 𝑋 → ( 0 < 𝑥0 < 𝑋))
98ifbid 4553 . . . 4 (𝑥 = 𝑋 → if( 0 < 𝑥, 1, -1) = if( 0 < 𝑋, 1, -1))
107, 9ifbieq2d 4556 . . 3 (𝑥 = 𝑋 → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
1110adantl 480 . 2 (((𝑅𝑉𝑋𝐵) ∧ 𝑥 = 𝑋) → if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
12 simpr 483 . 2 ((𝑅𝑉𝑋𝐵) → 𝑋𝐵)
13 c0ex 11240 . . . 4 0 ∈ V
1413a1i 11 . . 3 (((𝑅𝑉𝑋𝐵) ∧ 𝑋 = 0 ) → 0 ∈ V)
15 1ex 11242 . . . . 5 1 ∈ V
1615a1i 11 . . . 4 ((((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) ∧ 0 < 𝑋) → 1 ∈ V)
17 negex 11490 . . . . 5 -1 ∈ V
1817a1i 11 . . . 4 ((((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) ∧ ¬ 0 < 𝑋) → -1 ∈ V)
1916, 18ifclda 4565 . . 3 (((𝑅𝑉𝑋𝐵) ∧ ¬ 𝑋 = 0 ) → if( 0 < 𝑋, 1, -1) ∈ V)
2014, 19ifclda 4565 . 2 ((𝑅𝑉𝑋𝐵) → if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)) ∈ V)
216, 11, 12, 20fvmptd 7011 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 394   = wceq 1533  wcel 2098  Vcvv 3461  ifcif 4530   class class class wbr 5149  cmpt 5232  cfv 6549  0cc0 11140  1c1 11141  -cneg 11477  Basecbs 17183  0gc0g 17424  ltcplt 18303  sgnscsgns 32971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-mulcl 11202  ax-i2m1 11208
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-neg 11479  df-sgns 32972
This theorem is referenced by: (None)
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