Theorem dvrfvald 14112

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 14111	. . 3 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
2		fveq2 5629	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		fveq2 5629	. . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
4		fveq2 5629	. . . . 5 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
5		eqidd 2230	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥)
6		fveq2 5629	. . . . . 6 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅))
7	6	fveq1d 5631	. . . . 5 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = ((invr‘𝑅)‘𝑦))
8	4, 5, 7	oveq123d 6028	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦)))
9	2, 3, 8	mpoeq123dv 6072	. . 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 2813	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
12		basfn 13106	. . . . 5 ⊢ Base Fn V
13		funfvex 5646	. . . . . 6 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
14	13	funfni 5423	. . . . 5 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
15	12, 11, 14	sylancr 414	. . . 4 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
16		eqidd 2230	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
17		eqidd 2230	. . . . . 6 ⊢ (𝜑 → (Unit‘𝑅) = (Unit‘𝑅))
18	16, 17, 10	unitssd 14088	. . . . 5 ⊢ (𝜑 → (Unit‘𝑅) ⊆ (Base‘𝑅))
19	15, 18	ssexd 4224	. . . 4 ⊢ (𝜑 → (Unit‘𝑅) ∈ V)
20		mpoexga 6364	. . . 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 5730	. 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 2230	. . . 4 ⊢ (𝜑 → 𝑥 = 𝑥)
28		dvrfvald.i	. . . . 5 ⊢ (𝜑 → 𝐼 = (invr‘𝑅))
29	28	fveq1d 5631	. . . 4 ⊢ (𝜑 → (𝐼‘𝑦) = ((invr‘𝑅)‘𝑦))
30	26, 27, 29	oveq123d 6028	. . 3 ⊢ (𝜑 → (𝑥 · (𝐼‘𝑦)) = (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦)))
31	24, 25, 30	mpoeq123dv 6072	. 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 2272	1 ⊢ (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2799   Fn wfn 5313  ‘cfv 5318  (class class class)co 6007   ∈ cmpo 6009  Basecbs 13047  ._rcmulr 13126  SRingcsrg 13941  Unitcui 14065  inv_rcinvr 14099  /_rcdvr 14110

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-plusg 13138  df-mulr 13139  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-mgp 13899  df-srg 13942  df-dvdsr 14067  df-unit 14068  df-dvr 14111

This theorem is referenced by:  dvrvald  14113