Theorem sgnsv 33294

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 3469	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6856	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		sgnsval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	eqtr4di 2809	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6		fveq2 6856	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
7		sgnsval.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
8	6, 7	eqtr4di 2809	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
9	8	adantr 483	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (0g‘𝑟) = 0 )
10	9	eqeq2d 2767	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g‘𝑟) ↔ 𝑥 = 0 ))
11		fveq2 6856	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅))
12		sgnsval.l	. . . . . . . . . 10 ⊢ < = (lt‘𝑅)
13	11, 12	eqtr4di 2809	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = < )
14	13	adantr 483	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < )
15		eqidd 2757	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥)
16	9, 14, 15	breq123d 5108	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → ((0g‘𝑟)(lt‘𝑟)𝑥 ↔ 0 < 𝑥))
17	16	ifbid 4498	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1))
18	10, 17	ifbieq2d 4501	. . . . 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 5180	. . . 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 33293	. . . 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 7201	. . 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 2803	1 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1554   ∈ wcel 2136  Vcvv 3448  ifcif 4474   class class class wbr 5094   ↦ cmpt 5175  ‘cfv 6510  0cc0 11063  1c1 11064  -cneg 11405  Basecbs 17221  0_gc0g 17444  ltcplt 18316  sgn_scsgns 33292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-sgns 33293

This theorem is referenced by:  sgnsval  33295  sgnsf  33296