![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pws1 | Structured version Visualization version GIF version |
Description: Value of the ring unit 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 2795 | . . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
3 | 1, 2 | pwsval 16593 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
4 | 3 | fveq2d 6547 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r‘𝑌) = (1r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
5 | eqid 2795 | . . 3 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
6 | simpr 485 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉) | |
7 | fvexd 6558 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Scalar‘𝑅) ∈ V) | |
8 | fconst6g 6441 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝐼 × {𝑅}):𝐼⟶Ring) | |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {𝑅}):𝐼⟶Ring) |
10 | 5, 6, 7, 9 | prds1 19059 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (1r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
11 | fn0g 17706 | . . . . . 6 ⊢ 0g Fn V | |
12 | fnmgp 18936 | . . . . . 6 ⊢ mulGrp Fn V | |
13 | ssv 3916 | . . . . . . 7 ⊢ ran mulGrp ⊆ V | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran mulGrp ⊆ V) |
15 | fnco 6340 | . . . . . 6 ⊢ ((0g Fn V ∧ mulGrp Fn V ∧ ran mulGrp ⊆ V) → (0g ∘ mulGrp) Fn V) | |
16 | 11, 12, 14, 15 | mp3an12i 1457 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (0g ∘ mulGrp) Fn V) |
17 | df-ur 18947 | . . . . . 6 ⊢ 1r = (0g ∘ mulGrp) | |
18 | 17 | fneq1i 6325 | . . . . 5 ⊢ (1r Fn V ↔ (0g ∘ mulGrp) Fn V) |
19 | 16, 18 | sylibr 235 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 1r Fn V) |
20 | elex 3455 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V) | |
21 | 20 | adantr 481 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V) |
22 | fcoconst 6764 | . . . 4 ⊢ ((1r Fn V ∧ 𝑅 ∈ V) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × {(1r‘𝑅)})) | |
23 | 19, 21, 22 | syl2anc 584 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × {(1r‘𝑅)})) |
24 | pws1.o | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
25 | 24 | sneqi 4487 | . . . 4 ⊢ { 1 } = {(1r‘𝑅)} |
26 | 25 | xpeq2i 5475 | . . 3 ⊢ (𝐼 × { 1 }) = (𝐼 × {(1r‘𝑅)}) |
27 | 23, 26 | syl6eqr 2849 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × { 1 })) |
28 | 4, 10, 27 | 3eqtr2rd 2838 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 1 }) = (1r‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ⊆ wss 3863 {csn 4476 × cxp 5446 ran crn 5449 ∘ ccom 5452 Fn wfn 6225 ⟶wf 6226 ‘cfv 6230 (class class class)co 7021 Scalarcsca 16402 0gc0g 16547 Xscprds 16553 ↑s cpws 16554 mulGrpcmgp 18934 1rcur 18946 Ringcrg 18992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-ixp 8316 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-sup 8757 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-fz 12748 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-plusg 16412 df-mulr 16413 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-hom 16423 df-cco 16424 df-0g 16549 df-prds 16555 df-pws 16557 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-mgp 18935 df-ur 18947 df-ring 18994 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |