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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 2737 | . . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
3 | 1, 2 | pwsval 17327 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
4 | 3 | fveq2d 6843 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r‘𝑌) = (1r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
5 | eqid 2737 | . . 3 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
6 | simpr 485 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉) | |
7 | fvexd 6854 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Scalar‘𝑅) ∈ V) | |
8 | fconst6g 6728 | . . . 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 19990 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (1r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
11 | fn0g 18477 | . . . . . 6 ⊢ 0g Fn V | |
12 | fnmgp 19856 | . . . . . 6 ⊢ mulGrp Fn V | |
13 | ssv 3966 | . . . . . . 7 ⊢ ran mulGrp ⊆ V | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran mulGrp ⊆ V) |
15 | fnco 6615 | . . . . . 6 ⊢ ((0g Fn V ∧ mulGrp Fn V ∧ ran mulGrp ⊆ V) → (0g ∘ mulGrp) Fn V) | |
16 | 11, 12, 14, 15 | mp3an12i 1465 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (0g ∘ mulGrp) Fn V) |
17 | df-ur 19872 | . . . . . 6 ⊢ 1r = (0g ∘ mulGrp) | |
18 | 17 | fneq1i 6596 | . . . . 5 ⊢ (1r Fn V ↔ (0g ∘ mulGrp) Fn V) |
19 | 16, 18 | sylibr 233 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 1r Fn V) |
20 | elex 3461 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V) | |
21 | 20 | adantr 481 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V) |
22 | fcoconst 7076 | . . . 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 4595 | . . . 4 ⊢ { 1 } = {(1r‘𝑅)} |
26 | 25 | xpeq2i 5658 | . . 3 ⊢ (𝐼 × { 1 }) = (𝐼 × {(1r‘𝑅)}) |
27 | 23, 26 | eqtr4di 2795 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × { 1 })) |
28 | 4, 10, 27 | 3eqtr2rd 2784 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 1 }) = (1r‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ⊆ wss 3908 {csn 4584 × cxp 5629 ran crn 5632 ∘ ccom 5635 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 Scalarcsca 17095 0gc0g 17280 Xscprds 17286 ↑s cpws 17287 mulGrpcmgp 19854 1rcur 19871 Ringcrg 19917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-plusg 17105 df-mulr 17106 df-sca 17108 df-vsca 17109 df-ip 17110 df-tset 17111 df-ple 17112 df-ds 17114 df-hom 17116 df-cco 17117 df-0g 17282 df-prds 17288 df-pws 17290 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-mgp 19855 df-ur 19872 df-ring 19919 |
This theorem is referenced by: pwspjmhmmgpd 19995 evlsbagval 40663 |
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