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Mirrors > Home > MPE Home > Th. List > ressply1mul | Structured version Visualization version GIF version |
Description: A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.) |
Ref | Expression |
---|---|
ressply1.s | ⊢ 𝑆 = (Poly1‘𝑅) |
ressply1.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
ressply1.u | ⊢ 𝑈 = (Poly1‘𝐻) |
ressply1.b | ⊢ 𝐵 = (Base‘𝑈) |
ressply1.2 | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
ressply1.p | ⊢ 𝑃 = (𝑆 ↾s 𝐵) |
Ref | Expression |
---|---|
ressply1mul | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . . 3 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
2 | ressply1.h | . . 3 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
3 | eqid 2771 | . . 3 ⊢ (1𝑜 mPoly 𝐻) = (1𝑜 mPoly 𝐻) | |
4 | ressply1.u | . . . 4 ⊢ 𝑈 = (Poly1‘𝐻) | |
5 | eqid 2771 | . . . 4 ⊢ (PwSer1‘𝐻) = (PwSer1‘𝐻) | |
6 | ressply1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑈) | |
7 | 4, 5, 6 | ply1bas 19776 | . . 3 ⊢ 𝐵 = (Base‘(1𝑜 mPoly 𝐻)) |
8 | 1on 7720 | . . . 4 ⊢ 1𝑜 ∈ On | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 1𝑜 ∈ On) |
10 | ressply1.2 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
11 | eqid 2771 | . . 3 ⊢ ((1𝑜 mPoly 𝑅) ↾s 𝐵) = ((1𝑜 mPoly 𝑅) ↾s 𝐵) | |
12 | 1, 2, 3, 7, 9, 10, 11 | ressmplmul 19669 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘(1𝑜 mPoly 𝐻))𝑌) = (𝑋(.r‘((1𝑜 mPoly 𝑅) ↾s 𝐵))𝑌)) |
13 | eqid 2771 | . . . 4 ⊢ (.r‘𝑈) = (.r‘𝑈) | |
14 | 4, 3, 13 | ply1mulr 19808 | . . 3 ⊢ (.r‘𝑈) = (.r‘(1𝑜 mPoly 𝐻)) |
15 | 14 | oveqi 6805 | . 2 ⊢ (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘(1𝑜 mPoly 𝐻))𝑌) |
16 | ressply1.s | . . . . 5 ⊢ 𝑆 = (Poly1‘𝑅) | |
17 | eqid 2771 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
18 | 16, 1, 17 | ply1mulr 19808 | . . . 4 ⊢ (.r‘𝑆) = (.r‘(1𝑜 mPoly 𝑅)) |
19 | 6 | fvexi 6342 | . . . . 5 ⊢ 𝐵 ∈ V |
20 | ressply1.p | . . . . . 6 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
21 | 20, 17 | ressmulr 16210 | . . . . 5 ⊢ (𝐵 ∈ V → (.r‘𝑆) = (.r‘𝑃)) |
22 | 19, 21 | ax-mp 5 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑃) |
23 | eqid 2771 | . . . . . 6 ⊢ (.r‘(1𝑜 mPoly 𝑅)) = (.r‘(1𝑜 mPoly 𝑅)) | |
24 | 11, 23 | ressmulr 16210 | . . . . 5 ⊢ (𝐵 ∈ V → (.r‘(1𝑜 mPoly 𝑅)) = (.r‘((1𝑜 mPoly 𝑅) ↾s 𝐵))) |
25 | 19, 24 | ax-mp 5 | . . . 4 ⊢ (.r‘(1𝑜 mPoly 𝑅)) = (.r‘((1𝑜 mPoly 𝑅) ↾s 𝐵)) |
26 | 18, 22, 25 | 3eqtr3i 2801 | . . 3 ⊢ (.r‘𝑃) = (.r‘((1𝑜 mPoly 𝑅) ↾s 𝐵)) |
27 | 26 | oveqi 6805 | . 2 ⊢ (𝑋(.r‘𝑃)𝑌) = (𝑋(.r‘((1𝑜 mPoly 𝑅) ↾s 𝐵))𝑌) |
28 | 12, 15, 27 | 3eqtr4g 2830 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 Oncon0 5864 ‘cfv 6029 (class class class)co 6792 1𝑜c1o 7706 Basecbs 16060 ↾s cress 16061 .rcmulr 16146 SubRingcsubrg 18982 mPoly cmpl 19564 PwSer1cps1 19756 Poly1cpl1 19758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-of 7044 df-ofr 7045 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11496 df-z 11581 df-dec 11697 df-uz 11890 df-fz 12530 df-seq 13005 df-struct 16062 df-ndx 16063 df-slot 16064 df-base 16066 df-sets 16067 df-ress 16068 df-plusg 16158 df-mulr 16159 df-sca 16161 df-vsca 16162 df-tset 16164 df-ple 16165 df-0g 16306 df-gsum 16307 df-mgm 17446 df-sgrp 17488 df-mnd 17499 df-submnd 17540 df-grp 17629 df-minusg 17630 df-subg 17795 df-mgp 18694 df-ring 18753 df-subrg 18984 df-psr 19567 df-mpl 19569 df-opsr 19571 df-psr1 19761 df-ply1 19763 |
This theorem is referenced by: (None) |
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