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Theorem dvrfvald 13255
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 (𝜑𝐵 = (Base‘𝑅))
dvrfvald.t (𝜑· = (.r𝑅))
dvrfvald.u (𝜑𝑈 = (Unit‘𝑅))
dvrfvald.i (𝜑𝐼 = (invr𝑅))
dvrfvald.d (𝜑/ = (/r𝑅))
dvrfvald.r (𝜑𝑅 ∈ SRing)
Assertion
Ref Expression
dvrfvald (𝜑/ = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
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.
StepHypRef Expression
1 df-dvr 13254 . . 3 /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))))
2 fveq2 5515 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 fveq2 5515 . . . 4 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
4 fveq2 5515 . . . . 5 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
5 eqidd 2178 . . . . 5 (𝑟 = 𝑅𝑥 = 𝑥)
6 fveq2 5515 . . . . . 6 (𝑟 = 𝑅 → (invr𝑟) = (invr𝑅))
76fveq1d 5517 . . . . 5 (𝑟 = 𝑅 → ((invr𝑟)‘𝑦) = ((invr𝑅)‘𝑦))
84, 5, 7oveq123d 5895 . . . 4 (𝑟 = 𝑅 → (𝑥(.r𝑟)((invr𝑟)‘𝑦)) = (𝑥(.r𝑅)((invr𝑅)‘𝑦)))
92, 3, 8mpoeq123dv 5936 . . 3 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r𝑅)((invr𝑅)‘𝑦))))
10 dvrfvald.r . . . 4 (𝜑𝑅 ∈ SRing)
1110elexd 2750 . . 3 (𝜑𝑅 ∈ V)
12 basfn 12514 . . . . 5 Base Fn V
13 funfvex 5532 . . . . . 6 ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
1413funfni 5316 . . . . 5 ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
1512, 11, 14sylancr 414 . . . 4 (𝜑 → (Base‘𝑅) ∈ V)
16 eqidd 2178 . . . . . 6 (𝜑 → (Base‘𝑅) = (Base‘𝑅))
17 eqidd 2178 . . . . . 6 (𝜑 → (Unit‘𝑅) = (Unit‘𝑅))
1816, 17, 10unitssd 13231 . . . . 5 (𝜑 → (Unit‘𝑅) ⊆ (Base‘𝑅))
1915, 18ssexd 4143 . . . 4 (𝜑 → (Unit‘𝑅) ∈ V)
20 mpoexga 6212 . . . 4 (((Base‘𝑅) ∈ V ∧ (Unit‘𝑅) ∈ V) → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r𝑅)((invr𝑅)‘𝑦))) ∈ V)
2115, 19, 20syl2anc 411 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r𝑅)((invr𝑅)‘𝑦))) ∈ V)
221, 9, 11, 21fvmptd3 5609 . 2 (𝜑 → (/r𝑅) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r𝑅)((invr𝑅)‘𝑦))))
23 dvrfvald.d . 2 (𝜑/ = (/r𝑅))
24 dvrfvald.b . . 3 (𝜑𝐵 = (Base‘𝑅))
25 dvrfvald.u . . 3 (𝜑𝑈 = (Unit‘𝑅))
26 dvrfvald.t . . . 4 (𝜑· = (.r𝑅))
27 eqidd 2178 . . . 4 (𝜑𝑥 = 𝑥)
28 dvrfvald.i . . . . 5 (𝜑𝐼 = (invr𝑅))
2928fveq1d 5517 . . . 4 (𝜑 → (𝐼𝑦) = ((invr𝑅)‘𝑦))
3026, 27, 29oveq123d 5895 . . 3 (𝜑 → (𝑥 · (𝐼𝑦)) = (𝑥(.r𝑅)((invr𝑅)‘𝑦)))
3124, 25, 30mpoeq123dv 5936 . 2 (𝜑 → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Unit‘𝑅) ↦ (𝑥(.r𝑅)((invr𝑅)‘𝑦))))
3222, 23, 313eqtr4d 2220 1 (𝜑/ = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737   Fn wfn 5211  cfv 5216  (class class class)co 5874  cmpo 5876  Basecbs 12456  .rcmulr 12531  SRingcsrg 13099  Unitcui 13209  invrcinvr 13242  /rcdvr 13253
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 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-ltadd 7926
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-mgp 13084  df-srg 13100  df-dvdsr 13211  df-unit 13212  df-dvr 13254
This theorem is referenced by:  dvrvald  13256
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