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| Mirrors > Home > MPE Home > Th. List > ressmplmul | 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 |
|---|---|
| ressmpl.s | ⊢ 𝑆 = (𝐼 mPoly 𝑅) |
| ressmpl.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
| ressmpl.u | ⊢ 𝑈 = (𝐼 mPoly 𝐻) |
| ressmpl.b | ⊢ 𝐵 = (Base‘𝑈) |
| ressmpl.1 | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
| ressmpl.2 | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
| ressmpl.p | ⊢ 𝑃 = (𝑆 ↾s 𝐵) |
| Ref | Expression |
|---|---|
| ressmplmul | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ressmpl.u | . . . . . 6 ⊢ 𝑈 = (𝐼 mPoly 𝐻) | |
| 2 | eqid 2737 | . . . . . 6 ⊢ (𝐼 mPwSer 𝐻) = (𝐼 mPwSer 𝐻) | |
| 3 | ressmpl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑈) | |
| 4 | eqid 2737 | . . . . . 6 ⊢ (Base‘(𝐼 mPwSer 𝐻)) = (Base‘(𝐼 mPwSer 𝐻)) | |
| 5 | 1, 2, 3, 4 | mplbasss 22017 | . . . . 5 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝐻)) |
| 6 | 5 | sseli 3979 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘(𝐼 mPwSer 𝐻))) |
| 7 | 5 | sseli 3979 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ (Base‘(𝐼 mPwSer 𝐻))) |
| 8 | 6, 7 | anim12i 613 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ (Base‘(𝐼 mPwSer 𝐻)) ∧ 𝑌 ∈ (Base‘(𝐼 mPwSer 𝐻)))) |
| 9 | eqid 2737 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
| 10 | ressmpl.h | . . . 4 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
| 11 | eqid 2737 | . . . 4 ⊢ ((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))) = ((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))) | |
| 12 | ressmpl.2 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
| 13 | 9, 10, 2, 4, 11, 12 | resspsrmul 21996 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (Base‘(𝐼 mPwSer 𝐻)) ∧ 𝑌 ∈ (Base‘(𝐼 mPwSer 𝐻)))) → (𝑋(.r‘(𝐼 mPwSer 𝐻))𝑌) = (𝑋(.r‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))𝑌)) |
| 14 | 8, 13 | sylan2 593 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘(𝐼 mPwSer 𝐻))𝑌) = (𝑋(.r‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))𝑌)) |
| 15 | 3 | fvexi 6920 | . . . 4 ⊢ 𝐵 ∈ V |
| 16 | 1, 2, 3 | mplval2 22016 | . . . . 5 ⊢ 𝑈 = ((𝐼 mPwSer 𝐻) ↾s 𝐵) |
| 17 | eqid 2737 | . . . . 5 ⊢ (.r‘(𝐼 mPwSer 𝐻)) = (.r‘(𝐼 mPwSer 𝐻)) | |
| 18 | 16, 17 | ressmulr 17351 | . . . 4 ⊢ (𝐵 ∈ V → (.r‘(𝐼 mPwSer 𝐻)) = (.r‘𝑈)) |
| 19 | 15, 18 | ax-mp 5 | . . 3 ⊢ (.r‘(𝐼 mPwSer 𝐻)) = (.r‘𝑈) |
| 20 | 19 | oveqi 7444 | . 2 ⊢ (𝑋(.r‘(𝐼 mPwSer 𝐻))𝑌) = (𝑋(.r‘𝑈)𝑌) |
| 21 | fvex 6919 | . . . . 5 ⊢ (Base‘𝑆) ∈ V | |
| 22 | ressmpl.s | . . . . . . 7 ⊢ 𝑆 = (𝐼 mPoly 𝑅) | |
| 23 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
| 24 | 22, 9, 23 | mplval2 22016 | . . . . . 6 ⊢ 𝑆 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑆)) |
| 25 | eqid 2737 | . . . . . 6 ⊢ (.r‘(𝐼 mPwSer 𝑅)) = (.r‘(𝐼 mPwSer 𝑅)) | |
| 26 | 24, 25 | ressmulr 17351 | . . . . 5 ⊢ ((Base‘𝑆) ∈ V → (.r‘(𝐼 mPwSer 𝑅)) = (.r‘𝑆)) |
| 27 | 21, 26 | ax-mp 5 | . . . 4 ⊢ (.r‘(𝐼 mPwSer 𝑅)) = (.r‘𝑆) |
| 28 | fvex 6919 | . . . . 5 ⊢ (Base‘(𝐼 mPwSer 𝐻)) ∈ V | |
| 29 | 11, 25 | ressmulr 17351 | . . . . 5 ⊢ ((Base‘(𝐼 mPwSer 𝐻)) ∈ V → (.r‘(𝐼 mPwSer 𝑅)) = (.r‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))) |
| 30 | 28, 29 | ax-mp 5 | . . . 4 ⊢ (.r‘(𝐼 mPwSer 𝑅)) = (.r‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻)))) |
| 31 | ressmpl.p | . . . . . 6 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
| 32 | eqid 2737 | . . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
| 33 | 31, 32 | ressmulr 17351 | . . . . 5 ⊢ (𝐵 ∈ V → (.r‘𝑆) = (.r‘𝑃)) |
| 34 | 15, 33 | ax-mp 5 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑃) |
| 35 | 27, 30, 34 | 3eqtr3i 2773 | . . 3 ⊢ (.r‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻)))) = (.r‘𝑃) |
| 36 | 35 | oveqi 7444 | . 2 ⊢ (𝑋(.r‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))𝑌) = (𝑋(.r‘𝑃)𝑌) |
| 37 | 14, 20, 36 | 3eqtr3g 2800 | 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 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 .rcmulr 17298 SubRingcsubrg 20569 mPwSer cmps 21924 mPoly cmpl 21926 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-subrng 20546 df-subrg 20570 df-psr 21929 df-mpl 21931 |
| This theorem is referenced by: ressply1mul 22232 |
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