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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 2737 | . . . . 5 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
| 2 | eqid 2737 | . . . . 5 ⊢ ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) | |
| 3 | 1, 2 | unitgrpbas 20382 | . . . 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 20417 | . . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
| 9 | eqid 2737 | . . . . 5 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
| 10 | eqid 2737 | . . . . 5 ⊢ ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) | |
| 11 | 9, 10 | unitgrpbas 20382 | . . . 4 ⊢ (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
| 12 | 8, 11 | eqtrdi 2793 | . . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
| 13 | eqid 2737 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 14 | 13, 1 | unitss 20376 | . . . . . . . 8 ⊢ (Unit‘𝐾) ⊆ (Base‘𝐾) |
| 15 | 14, 5 | sseqtrrid 4027 | . . . . . . 7 ⊢ (𝜑 → (Unit‘𝐾) ⊆ 𝐵) |
| 16 | 15 | sselda 3983 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Unit‘𝐾)) → 𝑥 ∈ 𝐵) |
| 17 | 15 | sselda 3983 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Unit‘𝐾)) → 𝑦 ∈ 𝐵) |
| 18 | 16, 17 | anim12dan 619 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) |
| 19 | 18, 7 | syldan 591 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
| 20 | fvex 6919 | . . . . . 6 ⊢ (Unit‘𝐾) ∈ V | |
| 21 | eqid 2737 | . . . . . . . 8 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
| 22 | eqid 2737 | . . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
| 23 | 21, 22 | mgpplusg 20141 | . . . . . . 7 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
| 24 | 2, 23 | ressplusg 17334 | . . . . . 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 7444 | . . . 4 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦) |
| 27 | fvex 6919 | . . . . . 6 ⊢ (Unit‘𝐿) ∈ V | |
| 28 | eqid 2737 | . . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
| 29 | eqid 2737 | . . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
| 30 | 28, 29 | mgpplusg 20141 | . . . . . . 7 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
| 31 | 10, 30 | ressplusg 17334 | . . . . . 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 7444 | . . . 4 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))𝑦) |
| 34 | 19, 26, 33 | 3eqtr3g 2800 | . . 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 19033 | . 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
| 36 | eqid 2737 | . . 3 ⊢ (invr‘𝐾) = (invr‘𝐾) | |
| 37 | 1, 2, 36 | invrfval 20389 | . 2 ⊢ (invr‘𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) |
| 38 | eqid 2737 | . . 3 ⊢ (invr‘𝐿) = (invr‘𝐿) | |
| 39 | 9, 10, 38 | invrfval 20389 | . 2 ⊢ (invr‘𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
| 40 | 35, 37, 39 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 .rcmulr 17298 invgcminusg 18952 mulGrpcmgp 20137 Unitcui 20355 invrcinvr 20387 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-minusg 18955 df-mgp 20138 df-ur 20179 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 |
| This theorem is referenced by: (None) |
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