Theorem opprnegg 13259

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 2177	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3	1, 2	opprbasg 13253	. . 3 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
4		eqid 2177	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
5	1, 4	oppraddg 13254	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂))
6	5	oveqd 5895	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝑦(+g‘𝑅)𝑥) = (𝑦(+g‘𝑂)𝑥))
7		eqid 2177	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
8	1, 7	oppr0g 13257	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (0g‘𝑅) = (0g‘𝑂))
9	6, 8	eqeq12d 2192	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(+g‘𝑅)𝑥) = (0g‘𝑅) ↔ (𝑦(+g‘𝑂)𝑥) = (0g‘𝑂)))
10	3, 9	riotaeqbidv 5837	. . 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 4087	. 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 12921	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑁 = (𝑥 ∈ (Base‘𝑅) ↦ (℩𝑦 ∈ (Base‘𝑅)(𝑦(+g‘𝑅)𝑥) = (0g‘𝑅))))
14	1	opprex 13251	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
15		eqid 2177	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
16		eqid 2177	. . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂)
17		eqid 2177	. . . 4 ⊢ (0g‘𝑂) = (0g‘𝑂)
18		eqid 2177	. . . 4 ⊢ (invg‘𝑂) = (invg‘𝑂)
19	15, 16, 17, 18	grpinvfvalg 12921	. . 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 2220	1 ⊢ (𝑅 ∈ 𝑉 → 𝑁 = (invg‘𝑂))

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ↦ cmpt 4066  ‘cfv 5218  ℩crio 5833  (class class class)co 5878  Basecbs 12465  +_gcplusg 12539  0_gc0g 12711  inv_gcminusg 12884  opp_rcoppr 13245

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-pre-ltirr 7926  ax-pre-lttrn 7928  ax-pre-ltadd 7930

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-tpos 6249  df-pnf 7997  df-mnf 7998  df-ltxr 8000  df-inn 8923  df-2 8981  df-3 8982  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-plusg 12552  df-mulr 12553  df-0g 12713  df-minusg 12887  df-oppr 13246

This theorem is referenced by:  unitnegcl  13305