Theorem sgnsv 33119

Description: The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)

Hypotheses

Ref	Expression
sgnsval.b	⊢ 𝐵 = (Base‘𝑅)
sgnsval.0	⊢ 0 = (0g‘𝑅)
sgnsval.l	⊢ < = (lt‘𝑅)
sgnsval.s	⊢ 𝑆 = (sgns‘𝑅)

Assertion

Ref	Expression
sgnsv	⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ 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.

Step	Hyp	Ref	Expression
1		sgnsval.s	. 2 ⊢ 𝑆 = (sgns‘𝑅)
2		elex 3455	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6817	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		sgnsval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	eqtr4di 2783	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6		fveq2 6817	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
7		sgnsval.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
8	6, 7	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
9	8	adantr 480	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (0g‘𝑟) = 0 )
10	9	eqeq2d 2741	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g‘𝑟) ↔ 𝑥 = 0 ))
11		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅))
12		sgnsval.l	. . . . . . . . . 10 ⊢ < = (lt‘𝑅)
13	11, 12	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = < )
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < )
15		eqidd 2731	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥)
16	9, 14, 15	breq123d 5103	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → ((0g‘𝑟)(lt‘𝑟)𝑥 ↔ 0 < 𝑥))
17	16	ifbid 4497	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1))
18	10, 17	ifbieq2d 4500	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1)) = if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))
19	5, 18	mpteq12dva 5175	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1))) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
20		df-sgns 33118	. . . 4 ⊢ sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1))))
21	19, 20, 4	mptfvmpt 7157	. . 3 ⊢ (𝑅 ∈ V → (sgns‘𝑅) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
22	2, 21	syl 17	. 2 ⊢ (𝑅 ∈ 𝑉 → (sgns‘𝑅) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
23	1, 22	eqtrid 2777	1 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  Vcvv 3434  ifcif 4473   class class class wbr 5089   ↦ cmpt 5170  ‘cfv 6477  0cc0 10998  1c1 10999  -cneg 11337  Basecbs 17112  0_gc0g 17335  ltcplt 18206  sgn_scsgns 33117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-sgns 33118

This theorem is referenced by:  sgnsval  33120  sgnsf  33121