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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 2610 | . . . . 5 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
2 | eqid 2610 | . . . . 5 ⊢ ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) | |
3 | 1, 2 | unitgrpbas 18489 | . . . 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 18520 | . . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
9 | eqid 2610 | . . . . 5 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
10 | eqid 2610 | . . . . 5 ⊢ ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) | |
11 | 9, 10 | unitgrpbas 18489 | . . . 4 ⊢ (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
12 | 8, 11 | syl6eq 2660 | . . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
13 | eqid 2610 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | 13, 1 | unitss 18483 | . . . . . . . 8 ⊢ (Unit‘𝐾) ⊆ (Base‘𝐾) |
15 | 14, 5 | syl5sseqr 3617 | . . . . . . 7 ⊢ (𝜑 → (Unit‘𝐾) ⊆ 𝐵) |
16 | 15 | sselda 3568 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Unit‘𝐾)) → 𝑥 ∈ 𝐵) |
17 | 15 | sselda 3568 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Unit‘𝐾)) → 𝑦 ∈ 𝐵) |
18 | 16, 17 | anim12dan 878 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) |
19 | 18, 7 | syldan 486 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
20 | fvex 6113 | . . . . . 6 ⊢ (Unit‘𝐾) ∈ V | |
21 | eqid 2610 | . . . . . . . 8 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
22 | eqid 2610 | . . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
23 | 21, 22 | mgpplusg 18316 | . . . . . . 7 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
24 | 2, 23 | ressplusg 15818 | . . . . . 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 6562 | . . . 4 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦) |
27 | fvex 6113 | . . . . . 6 ⊢ (Unit‘𝐿) ∈ V | |
28 | eqid 2610 | . . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
29 | eqid 2610 | . . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
30 | 28, 29 | mgpplusg 18316 | . . . . . . 7 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
31 | 10, 30 | ressplusg 15818 | . . . . . 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 6562 | . . . 4 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))𝑦) |
34 | 19, 26, 33 | 3eqtr3g 2667 | . . 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 17313 | . 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
36 | eqid 2610 | . . 3 ⊢ (invr‘𝐾) = (invr‘𝐾) | |
37 | 1, 2, 36 | invrfval 18496 | . 2 ⊢ (invr‘𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) |
38 | eqid 2610 | . . 3 ⊢ (invr‘𝐿) = (invr‘𝐿) | |
39 | 9, 10, 38 | invrfval 18496 | . 2 ⊢ (invr‘𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
40 | 35, 37, 39 | 3eqtr4g 2669 | 1 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 +gcplusg 15768 .rcmulr 15769 invgcminusg 17246 mulGrpcmgp 18312 Unitcui 18462 invrcinvr 18494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-minusg 17249 df-mgp 18313 df-ur 18325 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 |
This theorem is referenced by: (None) |
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