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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 2733 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | ressply1.h | . . 3 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
3 | eqid 2733 | . . 3 ⊢ (1o mPoly 𝐻) = (1o mPoly 𝐻) | |
4 | ressply1.u | . . . 4 ⊢ 𝑈 = (Poly1‘𝐻) | |
5 | eqid 2733 | . . . 4 ⊢ (PwSer1‘𝐻) = (PwSer1‘𝐻) | |
6 | ressply1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑈) | |
7 | 4, 5, 6 | ply1bas 21711 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝐻)) |
8 | 1on 8475 | . . . 4 ⊢ 1o ∈ On | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ On) |
10 | ressply1.2 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
11 | eqid 2733 | . . 3 ⊢ ((1o mPoly 𝑅) ↾s 𝐵) = ((1o mPoly 𝑅) ↾s 𝐵) | |
12 | 1, 2, 3, 7, 9, 10, 11 | ressmplmul 21577 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘(1o mPoly 𝐻))𝑌) = (𝑋(.r‘((1o mPoly 𝑅) ↾s 𝐵))𝑌)) |
13 | eqid 2733 | . . . 4 ⊢ (.r‘𝑈) = (.r‘𝑈) | |
14 | 4, 3, 13 | ply1mulr 21741 | . . 3 ⊢ (.r‘𝑈) = (.r‘(1o mPoly 𝐻)) |
15 | 14 | oveqi 7419 | . 2 ⊢ (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘(1o mPoly 𝐻))𝑌) |
16 | ressply1.s | . . . . 5 ⊢ 𝑆 = (Poly1‘𝑅) | |
17 | eqid 2733 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
18 | 16, 1, 17 | ply1mulr 21741 | . . . 4 ⊢ (.r‘𝑆) = (.r‘(1o mPoly 𝑅)) |
19 | 6 | fvexi 6903 | . . . . 5 ⊢ 𝐵 ∈ V |
20 | ressply1.p | . . . . . 6 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
21 | 20, 17 | ressmulr 17249 | . . . . 5 ⊢ (𝐵 ∈ V → (.r‘𝑆) = (.r‘𝑃)) |
22 | 19, 21 | ax-mp 5 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑃) |
23 | eqid 2733 | . . . . . 6 ⊢ (.r‘(1o mPoly 𝑅)) = (.r‘(1o mPoly 𝑅)) | |
24 | 11, 23 | ressmulr 17249 | . . . . 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 2769 | . . 3 ⊢ (.r‘𝑃) = (.r‘((1o mPoly 𝑅) ↾s 𝐵)) |
27 | 26 | oveqi 7419 | . 2 ⊢ (𝑋(.r‘𝑃)𝑌) = (𝑋(.r‘((1o mPoly 𝑅) ↾s 𝐵))𝑌) |
28 | 12, 15, 27 | 3eqtr4g 2798 | 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 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 Oncon0 6362 ‘cfv 6541 (class class class)co 7406 1oc1o 8456 Basecbs 17141 ↾s cress 17170 .rcmulr 17195 SubRingcsubrg 20352 mPoly cmpl 21451 PwSer1cps1 21691 Poly1cpl1 21693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-seq 13964 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-tset 17213 df-ple 17214 df-0g 17384 df-gsum 17385 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-grp 18819 df-minusg 18820 df-subg 18998 df-mgp 19983 df-ring 20052 df-subrg 20354 df-psr 21454 df-mpl 21456 df-opsr 21458 df-psr1 21696 df-ply1 21698 |
This theorem is referenced by: evls1muld 32638 |
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