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 2821 | . . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
3 | 1, 2 | pwsval 16753 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
4 | 3 | fveq2d 6668 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r‘𝑌) = (1r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
5 | eqid 2821 | . . 3 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
6 | simpr 487 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉) | |
7 | fvexd 6679 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (Scalar‘𝑅) ∈ V) | |
8 | fconst6g 6562 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝐼 × {𝑅}):𝐼⟶Ring) | |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {𝑅}):𝐼⟶Ring) |
10 | 5, 6, 7, 9 | prds1 19358 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (1r‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
11 | fn0g 17867 | . . . . . 6 ⊢ 0g Fn V | |
12 | fnmgp 19235 | . . . . . 6 ⊢ mulGrp Fn V | |
13 | ssv 3990 | . . . . . . 7 ⊢ ran mulGrp ⊆ V | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran mulGrp ⊆ V) |
15 | fnco 6459 | . . . . . 6 ⊢ ((0g Fn V ∧ mulGrp Fn V ∧ ran mulGrp ⊆ V) → (0g ∘ mulGrp) Fn V) | |
16 | 11, 12, 14, 15 | mp3an12i 1461 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (0g ∘ mulGrp) Fn V) |
17 | df-ur 19246 | . . . . . 6 ⊢ 1r = (0g ∘ mulGrp) | |
18 | 17 | fneq1i 6444 | . . . . 5 ⊢ (1r Fn V ↔ (0g ∘ mulGrp) Fn V) |
19 | 16, 18 | sylibr 236 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 1r Fn V) |
20 | elex 3512 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V) | |
21 | 20 | adantr 483 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V) |
22 | fcoconst 6890 | . . . 4 ⊢ ((1r Fn V ∧ 𝑅 ∈ V) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × {(1r‘𝑅)})) | |
23 | 19, 21, 22 | syl2anc 586 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × {(1r‘𝑅)})) |
24 | pws1.o | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
25 | 24 | sneqi 4571 | . . . 4 ⊢ { 1 } = {(1r‘𝑅)} |
26 | 25 | xpeq2i 5576 | . . 3 ⊢ (𝐼 × { 1 }) = (𝐼 × {(1r‘𝑅)}) |
27 | 23, 26 | syl6eqr 2874 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (1r ∘ (𝐼 × {𝑅})) = (𝐼 × { 1 })) |
28 | 4, 10, 27 | 3eqtr2rd 2863 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 1 }) = (1r‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 {csn 4560 × cxp 5547 ran crn 5550 ∘ ccom 5553 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 Scalarcsca 16562 0gc0g 16707 Xscprds 16713 ↑s cpws 16714 mulGrpcmgp 19233 1rcur 19245 Ringcrg 19291 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-0g 16709 df-prds 16715 df-pws 16717 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mgp 19234 df-ur 19246 df-ring 19293 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |