Theorem dvrfvald 14278

Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)

Hypotheses

Ref	Expression
dvrfvald.b	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
dvrfvald.t	⊢ (𝜑 → · = (.r‘𝑅))
dvrfvald.u	⊢ (𝜑 → 𝑈 = (Unit‘𝑅))
dvrfvald.i	⊢ (𝜑 → 𝐼 = (invr‘𝑅))
dvrfvald.d	⊢ (𝜑 → / = (/r‘𝑅))
dvrfvald.r	⊢ (𝜑 → 𝑅 ∈ SRing)

Assertion

Ref	Expression
dvrfvald	⊢ (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐼,𝑦   𝑥,𝑅,𝑦   𝑥, · ,𝑦   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   / (𝑥,𝑦)

Proof of Theorem dvrfvald

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dvr 14277	. . 3 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
2		fveq2 5670	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		fveq2 5670	. . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
4		fveq2 5670	. . . . 5 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
5		eqidd 2233	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥)
6		fveq2 5670	. . . . . 6 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅))
7	6	fveq1d 5672	. . . . 5 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = ((invr‘𝑅)‘𝑦))
8	4, 5, 7	oveq123d 6071	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦)))
9	2, 3, 8	mpoeq123dv 6115	. . 3 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))))
10		dvrfvald.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ SRing)
11	10	elexd 2827	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
12		basfn 13271	. . . . 5 ⊢ Base Fn V
13		funfvex 5687	. . . . . 6 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
14	13	funfni 5458	. . . . 5 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
15	12, 11, 14	sylancr 414	. . . 4 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
16		eqidd 2233	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
17		eqidd 2233	. . . . . 6 ⊢ (𝜑 → (Unit‘𝑅) = (Unit‘𝑅))
18	16, 17, 10	unitssd 14254	. . . . 5 ⊢ (𝜑 → (Unit‘𝑅) ⊆ (Base‘𝑅))
19	15, 18	ssexd 4250	. . . 4 ⊢ (𝜑 → (Unit‘𝑅) ∈ V)
20		mpoexga 6408	. . . 4 ⊢ (((Base‘𝑅) ∈ V ∧ (Unit‘𝑅) ∈ V) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))) ∈ V)
21	15, 19, 20	syl2anc 411	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))) ∈ V)
22	1, 9, 11, 21	fvmptd3 5771	. 2 ⊢ (𝜑 → (/r‘𝑅) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))))
23		dvrfvald.d	. 2 ⊢ (𝜑 → / = (/r‘𝑅))
24		dvrfvald.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
25		dvrfvald.u	. . 3 ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))
26		dvrfvald.t	. . . 4 ⊢ (𝜑 → · = (.r‘𝑅))
27		eqidd 2233	. . . 4 ⊢ (𝜑 → 𝑥 = 𝑥)
28		dvrfvald.i	. . . . 5 ⊢ (𝜑 → 𝐼 = (invr‘𝑅))
29	28	fveq1d 5672	. . . 4 ⊢ (𝜑 → (𝐼‘𝑦) = ((invr‘𝑅)‘𝑦))
30	26, 27, 29	oveq123d 6071	. . 3 ⊢ (𝜑 → (𝑥 · (𝐼‘𝑦)) = (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦)))
31	24, 25, 30	mpoeq123dv 6115	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))))
32	22, 23, 31	3eqtr4d 2275	1 ⊢ (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2813   Fn wfn 5347  ‘cfv 5352  (class class class)co 6050   ∈ cmpo 6052  Basecbs 13212  ._rcmulr 13291  SRingcsrg 14107  Unitcui 14231  inv_rcinvr 14265  /_rcdvr 14276

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-mgp 14065  df-srg 14108  df-dvdsr 14233  df-unit 14234  df-dvr 14277

This theorem is referenced by:  dvrvald  14279