Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ply1ass23l | Structured version Visualization version GIF version |
Description: Associative identity with scalar and ring multiplication for the polynomial ring. (Contributed by AV, 14-Aug-2019.) |
Ref | Expression |
---|---|
ply1ass23l.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1ass23l.t | ⊢ × = (.r‘𝑃) |
ply1ass23l.b | ⊢ 𝐵 = (Base‘𝑃) |
ply1ass23l.k | ⊢ 𝐾 = (Base‘𝑅) |
ply1ass23l.n | ⊢ · = ( ·𝑠 ‘𝑃) |
Ref | Expression |
---|---|
ply1ass23l | ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . 2 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
2 | 1on 8198 | . . 3 ⊢ 1o ∈ On | |
3 | 2 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 1o ∈ On) |
4 | simpl 486 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑅 ∈ Ring) | |
5 | eqid 2734 | . 2 ⊢ {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
6 | eqid 2734 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
7 | ply1ass23l.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
8 | ply1ass23l.t | . . . 4 ⊢ × = (.r‘𝑃) | |
9 | 7, 6, 8 | ply1mulr 21120 | . . 3 ⊢ × = (.r‘(1o mPoly 𝑅)) |
10 | 6, 1, 9 | mplmulr 21114 | . 2 ⊢ × = (.r‘(1o mPwSer 𝑅)) |
11 | eqid 2734 | . 2 ⊢ (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅)) | |
12 | eqid 2734 | . . . . . 6 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
13 | 6, 1, 12, 11 | mplbasss 20931 | . . . . 5 ⊢ (Base‘(1o mPoly 𝑅)) ⊆ (Base‘(1o mPwSer 𝑅)) |
14 | ply1ass23l.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑃) | |
15 | 7, 14 | ply1bascl2 21097 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘(1o mPoly 𝑅))) |
16 | 13, 15 | sseldi 3889 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘(1o mPwSer 𝑅))) |
17 | 16 | 3ad2ant2 1136 | . . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ (Base‘(1o mPwSer 𝑅))) |
18 | 17 | adantl 485 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ (Base‘(1o mPwSer 𝑅))) |
19 | 7, 14 | ply1bascl2 21097 | . . . . 5 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ (Base‘(1o mPoly 𝑅))) |
20 | 13, 19 | sseldi 3889 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ (Base‘(1o mPwSer 𝑅))) |
21 | 20 | 3ad2ant3 1137 | . . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ (Base‘(1o mPwSer 𝑅))) |
22 | 21 | adantl 485 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ (Base‘(1o mPwSer 𝑅))) |
23 | ply1ass23l.k | . 2 ⊢ 𝐾 = (Base‘𝑅) | |
24 | ply1ass23l.n | . . . 4 ⊢ · = ( ·𝑠 ‘𝑃) | |
25 | 7, 6, 24 | ply1vsca 21119 | . . 3 ⊢ · = ( ·𝑠 ‘(1o mPoly 𝑅)) |
26 | 6, 1, 25 | mplvsca2 20946 | . 2 ⊢ · = ( ·𝑠 ‘(1o mPwSer 𝑅)) |
27 | simpr1 1196 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐴 ∈ 𝐾) | |
28 | 1, 3, 4, 5, 10, 11, 18, 22, 23, 26, 27 | psrass23l 20905 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 {crab 3058 ◡ccnv 5539 “ cima 5543 Oncon0 6202 ‘cfv 6369 (class class class)co 7202 1oc1o 8184 ↑m cmap 8497 Fincfn 8615 ℕcn 11813 ℕ0cn0 12073 Basecbs 16684 .rcmulr 16768 ·𝑠 cvsca 16771 Ringcrg 19534 mPwSer cmps 20835 mPoly cmpl 20837 Poly1cpl1 21070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-ofr 7459 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-tset 16786 df-ple 16787 df-0g 16918 df-gsum 16919 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-grp 18340 df-minusg 18341 df-ghm 18592 df-cntz 18683 df-cmn 19144 df-abl 19145 df-mgp 19477 df-ur 19489 df-ring 19536 df-psr 20840 df-mpl 20842 df-opsr 20844 df-psr1 21073 df-ply1 21075 |
This theorem is referenced by: ply1sclrmsm 45351 |
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