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Mirrors > Home > MPE Home > Th. List > invrpropd | Structured version Visualization version GIF version |
Description: The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
rngidpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
rngidpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
rngidpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
Ref | Expression |
---|---|
invrpropd | ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2739 | . . . . 5 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
2 | eqid 2739 | . . . . 5 ⊢ ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) | |
3 | 1, 2 | unitgrpbas 19889 | . . . 4 ⊢ (Unit‘𝐾) = (Base‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))) |
5 | rngidpropd.1 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
6 | rngidpropd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
7 | rngidpropd.3 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
8 | 5, 6, 7 | unitpropd 19920 | . . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
9 | eqid 2739 | . . . . 5 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
10 | eqid 2739 | . . . . 5 ⊢ ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) | |
11 | 9, 10 | unitgrpbas 19889 | . . . 4 ⊢ (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
12 | 8, 11 | eqtrdi 2795 | . . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
13 | eqid 2739 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | 13, 1 | unitss 19883 | . . . . . . . 8 ⊢ (Unit‘𝐾) ⊆ (Base‘𝐾) |
15 | 14, 5 | sseqtrrid 3978 | . . . . . . 7 ⊢ (𝜑 → (Unit‘𝐾) ⊆ 𝐵) |
16 | 15 | sselda 3925 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Unit‘𝐾)) → 𝑥 ∈ 𝐵) |
17 | 15 | sselda 3925 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Unit‘𝐾)) → 𝑦 ∈ 𝐵) |
18 | 16, 17 | anim12dan 618 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) |
19 | 18, 7 | syldan 590 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
20 | fvex 6781 | . . . . . 6 ⊢ (Unit‘𝐾) ∈ V | |
21 | eqid 2739 | . . . . . . . 8 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
22 | eqid 2739 | . . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
23 | 21, 22 | mgpplusg 19705 | . . . . . . 7 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
24 | 2, 23 | ressplusg 16981 | . . . . . 6 ⊢ ((Unit‘𝐾) ∈ V → (.r‘𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))) |
25 | 20, 24 | ax-mp 5 | . . . . 5 ⊢ (.r‘𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) |
26 | 25 | oveqi 7281 | . . . 4 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦) |
27 | fvex 6781 | . . . . . 6 ⊢ (Unit‘𝐿) ∈ V | |
28 | eqid 2739 | . . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
29 | eqid 2739 | . . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
30 | 28, 29 | mgpplusg 19705 | . . . . . . 7 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
31 | 10, 30 | ressplusg 16981 | . . . . . 6 ⊢ ((Unit‘𝐿) ∈ V → (.r‘𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
32 | 27, 31 | ax-mp 5 | . . . . 5 ⊢ (.r‘𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
33 | 32 | oveqi 7281 | . . . 4 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))𝑦) |
34 | 19, 26, 33 | 3eqtr3g 2802 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦) = (𝑥(+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))𝑦)) |
35 | 4, 12, 34 | grpinvpropd 18631 | . 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
36 | eqid 2739 | . . 3 ⊢ (invr‘𝐾) = (invr‘𝐾) | |
37 | 1, 2, 36 | invrfval 19896 | . 2 ⊢ (invr‘𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) |
38 | eqid 2739 | . . 3 ⊢ (invr‘𝐿) = (invr‘𝐿) | |
39 | 9, 10, 38 | invrfval 19896 | . 2 ⊢ (invr‘𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
40 | 35, 37, 39 | 3eqtr4g 2804 | 1 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 Vcvv 3430 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 ↾s cress 16922 +gcplusg 16943 .rcmulr 16944 invgcminusg 18559 mulGrpcmgp 19701 Unitcui 19862 invrcinvr 19894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-tpos 8026 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-0g 17133 df-minusg 18562 df-mgp 19702 df-ur 19719 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 |
This theorem is referenced by: (None) |
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