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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 2735 | . . . . 5 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
2 | eqid 2735 | . . . . 5 ⊢ ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) | |
3 | 1, 2 | unitgrpbas 20399 | . . . 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 20434 | . . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
9 | eqid 2735 | . . . . 5 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
10 | eqid 2735 | . . . . 5 ⊢ ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) | |
11 | 9, 10 | unitgrpbas 20399 | . . . 4 ⊢ (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
12 | 8, 11 | eqtrdi 2791 | . . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
13 | eqid 2735 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | 13, 1 | unitss 20393 | . . . . . . . 8 ⊢ (Unit‘𝐾) ⊆ (Base‘𝐾) |
15 | 14, 5 | sseqtrrid 4049 | . . . . . . 7 ⊢ (𝜑 → (Unit‘𝐾) ⊆ 𝐵) |
16 | 15 | sselda 3995 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Unit‘𝐾)) → 𝑥 ∈ 𝐵) |
17 | 15 | sselda 3995 | . . . . . 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 6920 | . . . . . 6 ⊢ (Unit‘𝐾) ∈ V | |
21 | eqid 2735 | . . . . . . . 8 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
22 | eqid 2735 | . . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
23 | 21, 22 | mgpplusg 20156 | . . . . . . 7 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
24 | 2, 23 | ressplusg 17336 | . . . . . 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 6920 | . . . . . 6 ⊢ (Unit‘𝐿) ∈ V | |
28 | eqid 2735 | . . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
29 | eqid 2735 | . . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
30 | 28, 29 | mgpplusg 20156 | . . . . . . 7 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
31 | 10, 30 | ressplusg 17336 | . . . . . 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 2798 | . . 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 19046 | . 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
36 | eqid 2735 | . . 3 ⊢ (invr‘𝐾) = (invr‘𝐾) | |
37 | 1, 2, 36 | invrfval 20406 | . 2 ⊢ (invr‘𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) |
38 | eqid 2735 | . . 3 ⊢ (invr‘𝐿) = (invr‘𝐿) | |
39 | 9, 10, 38 | invrfval 20406 | . 2 ⊢ (invr‘𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
40 | 35, 37, 39 | 3eqtr4g 2800 | 1 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 +gcplusg 17298 .rcmulr 17299 invgcminusg 18965 mulGrpcmgp 20152 Unitcui 20372 invrcinvr 20404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-minusg 18968 df-mgp 20153 df-ur 20200 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 |
This theorem is referenced by: (None) |
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