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Mirrors > Home > MPE Home > Th. List > pws1 | Structured version Visualization version GIF version |
Description: Value of the ring unity in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.) |
Ref | Expression |
---|---|
pws1.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pws1.o | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
pws1 | ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 1 }) = (1r‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pws1.y | . . . 4 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
2 | eqid 2726 | . . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
3 | 1, 2 | pwsval 17503 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
4 | 3 | fveq2d 6907 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r‘𝑌) = (1r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
5 | eqid 2726 | . . 3 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
6 | simpr 483 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉) | |
7 | fvexd 6918 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Scalar‘𝑅) ∈ V) | |
8 | fconst6g 6793 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝐼 × {𝑅}):𝐼⟶Ring) | |
9 | 8 | adantr 479 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {𝑅}):𝐼⟶Ring) |
10 | 5, 6, 7, 9 | prds1 20304 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (1r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
11 | fn0g 18658 | . . . . . 6 ⊢ 0g Fn V | |
12 | fnmgp 20121 | . . . . . 6 ⊢ mulGrp Fn V | |
13 | ssv 4004 | . . . . . . 7 ⊢ ran mulGrp ⊆ V | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran mulGrp ⊆ V) |
15 | fnco 6680 | . . . . . 6 ⊢ ((0g Fn V ∧ mulGrp Fn V ∧ ran mulGrp ⊆ V) → (0g ∘ mulGrp) Fn V) | |
16 | 11, 12, 14, 15 | mp3an12i 1462 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (0g ∘ mulGrp) Fn V) |
17 | df-ur 20167 | . . . . . 6 ⊢ 1r = (0g ∘ mulGrp) | |
18 | 17 | fneq1i 6659 | . . . . 5 ⊢ (1r Fn V ↔ (0g ∘ mulGrp) Fn V) |
19 | 16, 18 | sylibr 233 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 1r Fn V) |
20 | elex 3482 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V) | |
21 | 20 | adantr 479 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V) |
22 | fcoconst 7150 | . . . 4 ⊢ ((1r Fn V ∧ 𝑅 ∈ V) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × {(1r‘𝑅)})) | |
23 | 19, 21, 22 | syl2anc 582 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × {(1r‘𝑅)})) |
24 | pws1.o | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
25 | 24 | sneqi 4644 | . . . 4 ⊢ { 1 } = {(1r‘𝑅)} |
26 | 25 | xpeq2i 5711 | . . 3 ⊢ (𝐼 × { 1 }) = (𝐼 × {(1r‘𝑅)}) |
27 | 23, 26 | eqtr4di 2784 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × { 1 })) |
28 | 4, 10, 27 | 3eqtr2rd 2773 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 1 }) = (1r‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3947 {csn 4633 × cxp 5682 ran crn 5685 ∘ ccom 5688 Fn wfn 6551 ⟶wf 6552 ‘cfv 6556 (class class class)co 7426 Scalarcsca 17271 0gc0g 17456 Xscprds 17462 ↑s cpws 17463 mulGrpcmgp 20119 1rcur 20166 Ringcrg 20218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-map 8859 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9487 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12613 df-dec 12732 df-uz 12877 df-fz 13541 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-plusg 17281 df-mulr 17282 df-sca 17284 df-vsca 17285 df-ip 17286 df-tset 17287 df-ple 17288 df-ds 17290 df-hom 17292 df-cco 17293 df-0g 17458 df-prds 17464 df-pws 17466 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-mgp 20120 df-ur 20167 df-ring 20220 |
This theorem is referenced by: pwspjmhmmgpd 20309 evlsvvval 42233 |
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