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Mirrors > Home > ILE Home > Th. List > dvrfvald | GIF version |
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.) |
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) |
Ref | Expression |
---|---|
dvrfvald | ⊢ (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dvr 13254 | . . 3 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘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 → (.r‘𝑟) = (.r‘𝑅)) | |
5 | eqidd 2178 | . . . . 5 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥) | |
6 | fveq2 5515 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅)) | |
7 | 6 | fveq1d 5517 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = ((invr‘𝑅)‘𝑦)) |
8 | 4, 5, 7 | oveq123d 5895 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))) |
9 | 2, 3, 8 | mpoeq123dv 5936 | . . 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 2750 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
12 | basfn 12514 | . . . . 5 ⊢ Base Fn V | |
13 | funfvex 5532 | . . . . . 6 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V) | |
14 | 13 | funfni 5316 | . . . . 5 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V) |
15 | 12, 11, 14 | sylancr 414 | . . . 4 ⊢ (𝜑 → (Base‘𝑅) ∈ V) |
16 | eqidd 2178 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅)) | |
17 | eqidd 2178 | . . . . . 6 ⊢ (𝜑 → (Unit‘𝑅) = (Unit‘𝑅)) | |
18 | 16, 17, 10 | unitssd 13231 | . . . . 5 ⊢ (𝜑 → (Unit‘𝑅) ⊆ (Base‘𝑅)) |
19 | 15, 18 | ssexd 4143 | . . . 4 ⊢ (𝜑 → (Unit‘𝑅) ∈ V) |
20 | mpoexga 6212 | . . . 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 5609 | . 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 2178 | . . . 4 ⊢ (𝜑 → 𝑥 = 𝑥) | |
28 | dvrfvald.i | . . . . 5 ⊢ (𝜑 → 𝐼 = (invr‘𝑅)) | |
29 | 28 | fveq1d 5517 | . . . 4 ⊢ (𝜑 → (𝐼‘𝑦) = ((invr‘𝑅)‘𝑦)) |
30 | 26, 27, 29 | oveq123d 5895 | . . 3 ⊢ (𝜑 → (𝑥 · (𝐼‘𝑦)) = (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))) |
31 | 24, 25, 30 | mpoeq123dv 5936 | . 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 2220 | 1 ⊢ (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
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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