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Theorem sgnsv 32058
Description: The sign mapping. (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
sgnsv (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
Distinct variable groups:   𝑥, 0   𝑥, <   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem sgnsv
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 sgnsval.s . 2 𝑆 = (sgns𝑅)
2 elex 3462 . . 3 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6843 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 sgnsval.b . . . . . 6 𝐵 = (Base‘𝑅)
53, 4eqtr4di 2791 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6 fveq2 6843 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
7 sgnsval.0 . . . . . . . . 9 0 = (0g𝑅)
86, 7eqtr4di 2791 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
98adantr 482 . . . . . . 7 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (0g𝑟) = 0 )
109eqeq2d 2744 . . . . . 6 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g𝑟) ↔ 𝑥 = 0 ))
11 fveq2 6843 . . . . . . . . . 10 (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅))
12 sgnsval.l . . . . . . . . . 10 < = (lt‘𝑅)
1311, 12eqtr4di 2791 . . . . . . . . 9 (𝑟 = 𝑅 → (lt‘𝑟) = < )
1413adantr 482 . . . . . . . 8 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < )
15 eqidd 2734 . . . . . . . 8 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥)
169, 14, 15breq123d 5120 . . . . . . 7 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → ((0g𝑟)(lt‘𝑟)𝑥0 < 𝑥))
1716ifbid 4510 . . . . . 6 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → if((0g𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1))
1810, 17ifbieq2d 4513 . . . . 5 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1)) = if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))
195, 18mpteq12dva 5195 . . . 4 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))) = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
20 df-sgns 32057 . . . 4 sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))))
2119, 20, 4mptfvmpt 7179 . . 3 (𝑅 ∈ V → (sgns𝑅) = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
222, 21syl 17 . 2 (𝑅𝑉 → (sgns𝑅) = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
231, 22eqtrid 2785 1 (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3444  ifcif 4487   class class class wbr 5106  cmpt 5189  cfv 6497  0cc0 11056  1c1 11057  -cneg 11391  Basecbs 17088  0gc0g 17326  ltcplt 18202  sgnscsgns 32056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-sgns 32057
This theorem is referenced by:  sgnsval  32059  sgnsf  32060
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