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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 2735 | . . . . . 6 ⊢ (𝐼 mPwSer 𝐻) = (𝐼 mPwSer 𝐻) | |
3 | ressmpl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑈) | |
4 | eqid 2735 | . . . . . 6 ⊢ (Base‘(𝐼 mPwSer 𝐻)) = (Base‘(𝐼 mPwSer 𝐻)) | |
5 | 1, 2, 3, 4 | mplbasss 22035 | . . . . 5 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝐻)) |
6 | 5 | sseli 3991 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘(𝐼 mPwSer 𝐻))) |
7 | 5 | sseli 3991 | . . . 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 2735 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
10 | ressmpl.h | . . . 4 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
11 | eqid 2735 | . . . 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 22014 | . . 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 6921 | . . . 4 ⊢ 𝐵 ∈ V |
16 | 1, 2, 3 | mplval2 22034 | . . . . 5 ⊢ 𝑈 = ((𝐼 mPwSer 𝐻) ↾s 𝐵) |
17 | eqid 2735 | . . . . 5 ⊢ (.r‘(𝐼 mPwSer 𝐻)) = (.r‘(𝐼 mPwSer 𝐻)) | |
18 | 16, 17 | ressmulr 17353 | . . . 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 6920 | . . . . 5 ⊢ (Base‘𝑆) ∈ V | |
22 | ressmpl.s | . . . . . . 7 ⊢ 𝑆 = (𝐼 mPoly 𝑅) | |
23 | eqid 2735 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
24 | 22, 9, 23 | mplval2 22034 | . . . . . 6 ⊢ 𝑆 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑆)) |
25 | eqid 2735 | . . . . . 6 ⊢ (.r‘(𝐼 mPwSer 𝑅)) = (.r‘(𝐼 mPwSer 𝑅)) | |
26 | 24, 25 | ressmulr 17353 | . . . . 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 6920 | . . . . 5 ⊢ (Base‘(𝐼 mPwSer 𝐻)) ∈ V | |
29 | 11, 25 | ressmulr 17353 | . . . . 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 2735 | . . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
33 | 31, 32 | ressmulr 17353 | . . . . 5 ⊢ (𝐵 ∈ V → (.r‘𝑆) = (.r‘𝑃)) |
34 | 15, 33 | ax-mp 5 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑃) |
35 | 27, 30, 34 | 3eqtr3i 2771 | . . 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 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 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 SubRingcsubrg 20586 mPwSer cmps 21942 mPoly cmpl 21944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-seq 14040 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-tset 17317 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-subrng 20563 df-subrg 20587 df-psr 21947 df-mpl 21949 |
This theorem is referenced by: ressply1mul 22248 |
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