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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 2610 | . . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
3 | 1, 2 | pwsval 15969 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
4 | 3 | fveq2d 6107 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r‘𝑌) = (1r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
5 | eqid 2610 | . . 3 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
6 | simpr 476 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉) | |
7 | fvex 6113 | . . . 4 ⊢ (Scalar‘𝑅) ∈ V | |
8 | 7 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Scalar‘𝑅) ∈ V) |
9 | fconst6g 6007 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝐼 × {𝑅}):𝐼⟶Ring) | |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {𝑅}):𝐼⟶Ring) |
11 | 5, 6, 8, 10 | prds1 18437 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (1r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
12 | fn0g 17085 | . . . . . . 7 ⊢ 0g Fn V | |
13 | 12 | a1i 11 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 0g Fn V) |
14 | fnmgp 18314 | . . . . . . 7 ⊢ mulGrp Fn V | |
15 | 14 | a1i 11 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → mulGrp Fn V) |
16 | ssv 3588 | . . . . . . 7 ⊢ ran mulGrp ⊆ V | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran mulGrp ⊆ V) |
18 | fnco 5913 | . . . . . 6 ⊢ ((0g Fn V ∧ mulGrp Fn V ∧ ran mulGrp ⊆ V) → (0g ∘ mulGrp) Fn V) | |
19 | 13, 15, 17, 18 | syl3anc 1318 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (0g ∘ mulGrp) Fn V) |
20 | df-ur 18325 | . . . . . 6 ⊢ 1r = (0g ∘ mulGrp) | |
21 | 20 | fneq1i 5899 | . . . . 5 ⊢ (1r Fn V ↔ (0g ∘ mulGrp) Fn V) |
22 | 19, 21 | sylibr 223 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 1r Fn V) |
23 | elex 3185 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V) | |
24 | 23 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V) |
25 | fcoconst 6307 | . . . 4 ⊢ ((1r Fn V ∧ 𝑅 ∈ V) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × {(1r‘𝑅)})) | |
26 | 22, 24, 25 | syl2anc 691 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × {(1r‘𝑅)})) |
27 | pws1.o | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
28 | 27 | sneqi 4136 | . . . 4 ⊢ { 1 } = {(1r‘𝑅)} |
29 | 28 | xpeq2i 5060 | . . 3 ⊢ (𝐼 × { 1 }) = (𝐼 × {(1r‘𝑅)}) |
30 | 26, 29 | syl6eqr 2662 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × { 1 })) |
31 | 4, 11, 30 | 3eqtr2rd 2651 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 1 }) = (1r‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {csn 4125 × cxp 5036 ran crn 5039 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Scalarcsca 15771 0gc0g 15923 Xscprds 15929 ↑s cpws 15930 mulGrpcmgp 18312 1rcur 18324 Ringcrg 18370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-prds 15931 df-pws 15933 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mgp 18313 df-ur 18325 df-ring 18372 |
This theorem is referenced by: (None) |
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