Theorem opprnegg 14244

Description: The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
opprbas.1	⊢ 𝑂 = (oppr‘𝑅)
opprneg.2	⊢ 𝑁 = (invg‘𝑅)

Assertion

Ref	Expression
opprnegg	⊢ (𝑅 ∈ 𝑉 → 𝑁 = (invg‘𝑂))

Proof of Theorem opprnegg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprbas.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
2		eqid 2234	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	opprbasg 14236	. . 3 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
4		eqid 2234	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
5	1, 4	oppraddg 14237	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
6	5	oveqd 6069	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑦(+g‘𝑅)𝑥) = (𝑦(+g‘𝑂)𝑥))
7		eqid 2234	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
8	1, 7	oppr0g 14242	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (0g‘𝑅) = (0g‘𝑂))
9	6, 8	eqeq12d 2249	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(+g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦(+g‘𝑂)𝑥) = (0g‘𝑂)))
10	3, 9	riotaeqbidv 6008	. . 3 ⊢ (𝑅 ∈ 𝑉 → (℩𝑦 ∈ (Base‘𝑅)(𝑦(+g‘𝑅)𝑥) = (0g‘𝑅)) = (℩𝑦 ∈ (Base‘𝑂)(𝑦(+g‘𝑂)𝑥) = (0g‘𝑂)))
11	3, 10	mpteq12dv 4194	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑅) ↦ (℩𝑦 ∈ (Base‘𝑅)(𝑦(+g‘𝑅)𝑥) = (0g‘𝑅))) = (𝑥 ∈ (Base‘𝑂) ↦ (℩𝑦 ∈ (Base‘𝑂)(𝑦(+g‘𝑂)𝑥) = (0g‘𝑂))))
12		opprneg.2	. . 3 ⊢ 𝑁 = (invg‘𝑅)
13	2, 4, 7, 12	grpinvfvalg 13772	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑁 = (𝑥 ∈ (Base‘𝑅) ↦ (℩𝑦 ∈ (Base‘𝑅)(𝑦(+g‘𝑅)𝑥) = (0g‘𝑅))))
14	1	opprex 14234	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
15		eqid 2234	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
16		eqid 2234	. . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂)
17		eqid 2234	. . . 4 ⊢ (0g‘𝑂) = (0g‘𝑂)
18		eqid 2234	. . . 4 ⊢ (invg‘𝑂) = (invg‘𝑂)
19	15, 16, 17, 18	grpinvfvalg 13772	. . 3 ⊢ (𝑂 ∈ V → (invg‘𝑂) = (𝑥 ∈ (Base‘𝑂) ↦ (℩𝑦 ∈ (Base‘𝑂)(𝑦(+g‘𝑂)𝑥) = (0g‘𝑂))))
20	14, 19	syl 14	. 2 ⊢ (𝑅 ∈ 𝑉 → (invg‘𝑂) = (𝑥 ∈ (Base‘𝑂) ↦ (℩𝑦 ∈ (Base‘𝑂)(𝑦(+g‘𝑂)𝑥) = (0g‘𝑂))))
21	11, 13, 20	3eqtr4d 2277	1 ⊢ (𝑅 ∈ 𝑉 → 𝑁 = (invg‘𝑂))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ↦ cmpt 4173  ‘cfv 5354  ℩crio 6004  (class class class)co 6052  Basecbs 13229  +_gcplusg 13307  0_gc0g 13486  inv_gcminusg 13731  opp_rcoppr 14228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-tpos 6478  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-plusg 13320  df-mulr 13321  df-0g 13488  df-minusg 13734  df-oppr 14229

This theorem is referenced by:  unitnegcl  14292