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Mirrors > Home > MPE Home > Th. List > ply1idvr1 | Structured version Visualization version GIF version |
Description: The identity of a polynomial ring expressed as power of the polynomial variable. (Contributed by AV, 14-Aug-2019.) |
Ref | Expression |
---|---|
ply1idvr1.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1idvr1.x | ⊢ 𝑋 = (var1‘𝑅) |
ply1idvr1.n | ⊢ 𝑁 = (mulGrp‘𝑃) |
ply1idvr1.e | ⊢ ↑ = (.g‘𝑁) |
Ref | Expression |
---|---|
ply1idvr1 | ⊢ (𝑅 ∈ Ring → (0 ↑ 𝑋) = (1r‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2820 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2820 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | 1, 2 | ringidcl 19313 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
4 | ply1idvr1.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | ply1idvr1.x | . . . . 5 ⊢ 𝑋 = (var1‘𝑅) | |
6 | eqid 2820 | . . . . 5 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
7 | ply1idvr1.n | . . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃) | |
8 | ply1idvr1.e | . . . . 5 ⊢ ↑ = (.g‘𝑁) | |
9 | eqid 2820 | . . . . 5 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
10 | 1, 4, 5, 6, 7, 8, 9 | ply1scltm 20444 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → ((algSc‘𝑃)‘(1r‘𝑅)) = ((1r‘𝑅)( ·𝑠 ‘𝑃)(0 ↑ 𝑋))) |
11 | 3, 10 | mpdan 685 | . . 3 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘(1r‘𝑅)) = ((1r‘𝑅)( ·𝑠 ‘𝑃)(0 ↑ 𝑋))) |
12 | 4 | ply1sca 20416 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
13 | 12 | fveq2d 6667 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘(Scalar‘𝑃))) |
14 | 13 | oveq1d 7164 | . . 3 ⊢ (𝑅 ∈ Ring → ((1r‘𝑅)( ·𝑠 ‘𝑃)(0 ↑ 𝑋)) = ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0 ↑ 𝑋))) |
15 | 4 | ply1lmod 20415 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
16 | 0nn0 11906 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
17 | eqid 2820 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
18 | 4, 5, 7, 8, 17 | ply1moncl 20434 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ ℕ0) → (0 ↑ 𝑋) ∈ (Base‘𝑃)) |
19 | 16, 18 | mpan2 689 | . . . 4 ⊢ (𝑅 ∈ Ring → (0 ↑ 𝑋) ∈ (Base‘𝑃)) |
20 | eqid 2820 | . . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
21 | eqid 2820 | . . . . 5 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃)) | |
22 | 17, 20, 6, 21 | lmodvs1 19657 | . . . 4 ⊢ ((𝑃 ∈ LMod ∧ (0 ↑ 𝑋) ∈ (Base‘𝑃)) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0 ↑ 𝑋)) = (0 ↑ 𝑋)) |
23 | 15, 19, 22 | syl2anc 586 | . . 3 ⊢ (𝑅 ∈ Ring → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0 ↑ 𝑋)) = (0 ↑ 𝑋)) |
24 | 11, 14, 23 | 3eqtrrd 2860 | . 2 ⊢ (𝑅 ∈ Ring → (0 ↑ 𝑋) = ((algSc‘𝑃)‘(1r‘𝑅))) |
25 | eqid 2820 | . . 3 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
26 | 4, 9, 2, 25 | ply1scl1 20455 | . 2 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘(1r‘𝑅)) = (1r‘𝑃)) |
27 | 24, 26 | eqtrd 2855 | 1 ⊢ (𝑅 ∈ Ring → (0 ↑ 𝑋) = (1r‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 0cc0 10530 ℕ0cn0 11891 Basecbs 16478 Scalarcsca 16563 ·𝑠 cvsca 16564 .gcmg 18219 mulGrpcmgp 19234 1rcur 19246 Ringcrg 19292 LModclmod 19629 algSccascl 20079 var1cv1 20339 Poly1cpl1 20340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-ofr 7403 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 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 12890 df-fzo 13031 df-seq 13367 df-hash 13688 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-tset 16579 df-ple 16580 df-0g 16710 df-gsum 16711 df-mre 16852 df-mrc 16853 df-acs 16855 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-mhm 17951 df-submnd 17952 df-grp 18101 df-minusg 18102 df-sbg 18103 df-mulg 18220 df-subg 18271 df-ghm 18351 df-cntz 18442 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-ring 19294 df-subrg 19528 df-lmod 19631 df-lss 19699 df-ascl 20082 df-psr 20131 df-mvr 20132 df-mpl 20133 df-opsr 20135 df-psr1 20343 df-vr1 20344 df-ply1 20345 |
This theorem is referenced by: decpmatid 21373 pmatcollpwscmatlem1 21392 idpm2idmp 21404 |
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