Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 2824 | . 2 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
2 | 1on 8112 | . . 3 ⊢ 1o ∈ On | |
3 | 2 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 1o ∈ On) |
4 | simpl 485 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑅 ∈ Ring) | |
5 | eqid 2824 | . 2 ⊢ {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
6 | eqid 2824 | . . 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 20398 | . . 3 ⊢ × = (.r‘(1o mPoly 𝑅)) |
10 | 6, 1, 9 | mplmulr 20392 | . 2 ⊢ × = (.r‘(1o mPwSer 𝑅)) |
11 | eqid 2824 | . 2 ⊢ (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅)) | |
12 | eqid 2824 | . . . . . 6 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
13 | 6, 1, 12, 11 | mplbasss 20215 | . . . . 5 ⊢ (Base‘(1o mPoly 𝑅)) ⊆ (Base‘(1o mPwSer 𝑅)) |
14 | ply1ass23l.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑃) | |
15 | 7, 14 | ply1bascl2 20375 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘(1o mPoly 𝑅))) |
16 | 13, 15 | sseldi 3968 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘(1o mPwSer 𝑅))) |
17 | 16 | 3ad2ant2 1130 | . . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ (Base‘(1o mPwSer 𝑅))) |
18 | 17 | adantl 484 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ (Base‘(1o mPwSer 𝑅))) |
19 | 7, 14 | ply1bascl2 20375 | . . . . 5 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ (Base‘(1o mPoly 𝑅))) |
20 | 13, 19 | sseldi 3968 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ (Base‘(1o mPwSer 𝑅))) |
21 | 20 | 3ad2ant3 1131 | . . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ (Base‘(1o mPwSer 𝑅))) |
22 | 21 | adantl 484 | . 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 20397 | . . 3 ⊢ · = ( ·𝑠 ‘(1o mPoly 𝑅)) |
26 | 6, 1, 25 | mplvsca2 20229 | . 2 ⊢ · = ( ·𝑠 ‘(1o mPwSer 𝑅)) |
27 | simpr1 1190 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐴 ∈ 𝐾) | |
28 | 1, 3, 4, 5, 10, 11, 18, 22, 23, 26, 27 | psrass23l 20191 | 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 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {crab 3145 ◡ccnv 5557 “ cima 5561 Oncon0 6194 ‘cfv 6358 (class class class)co 7159 1oc1o 8098 ↑m cmap 8409 Fincfn 8512 ℕcn 11641 ℕ0cn0 11900 Basecbs 16486 .rcmulr 16569 ·𝑠 cvsca 16572 Ringcrg 19300 mPwSer cmps 20134 mPoly cmpl 20136 Poly1cpl1 20348 |
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 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-ofr 7413 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-tset 16587 df-ple 16588 df-0g 16718 df-gsum 16719 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-grp 18109 df-minusg 18110 df-ghm 18359 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-psr 20139 df-mpl 20141 df-opsr 20143 df-psr1 20351 df-ply1 20353 |
This theorem is referenced by: ply1sclrmsm 44444 |
Copyright terms: Public domain | W3C validator |