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Mirrors > Home > MPE Home > Th. List > ressmpladd | Structured version Visualization version GIF version |
Description: A restricted polynomial algebra has the same addition 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 |
---|---|
ressmpladd | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(+g‘𝑈)𝑌) = (𝑋(+g‘𝑃)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressmpl.u | . . . . . 6 ⊢ 𝑈 = (𝐼 mPoly 𝐻) | |
2 | eqid 2740 | . . . . . 6 ⊢ (𝐼 mPwSer 𝐻) = (𝐼 mPwSer 𝐻) | |
3 | ressmpl.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑈) | |
4 | eqid 2740 | . . . . . 6 ⊢ (Base‘(𝐼 mPwSer 𝐻)) = (Base‘(𝐼 mPwSer 𝐻)) | |
5 | 1, 2, 3, 4 | mplbasss 21201 | . . . . 5 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝐻)) |
6 | 5 | sseli 3922 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘(𝐼 mPwSer 𝐻))) |
7 | 5 | sseli 3922 | . . . 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 2740 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
10 | ressmpl.h | . . . 4 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
11 | eqid 2740 | . . . 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 | resspsradd 21183 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (Base‘(𝐼 mPwSer 𝐻)) ∧ 𝑌 ∈ (Base‘(𝐼 mPwSer 𝐻)))) → (𝑋(+g‘(𝐼 mPwSer 𝐻))𝑌) = (𝑋(+g‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))𝑌)) |
14 | 8, 13 | sylan2 593 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(+g‘(𝐼 mPwSer 𝐻))𝑌) = (𝑋(+g‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))𝑌)) |
15 | 3 | fvexi 6785 | . . . 4 ⊢ 𝐵 ∈ V |
16 | 1, 2, 3 | mplval2 21200 | . . . . 5 ⊢ 𝑈 = ((𝐼 mPwSer 𝐻) ↾s 𝐵) |
17 | eqid 2740 | . . . . 5 ⊢ (+g‘(𝐼 mPwSer 𝐻)) = (+g‘(𝐼 mPwSer 𝐻)) | |
18 | 16, 17 | ressplusg 16998 | . . . 4 ⊢ (𝐵 ∈ V → (+g‘(𝐼 mPwSer 𝐻)) = (+g‘𝑈)) |
19 | 15, 18 | ax-mp 5 | . . 3 ⊢ (+g‘(𝐼 mPwSer 𝐻)) = (+g‘𝑈) |
20 | 19 | oveqi 7284 | . 2 ⊢ (𝑋(+g‘(𝐼 mPwSer 𝐻))𝑌) = (𝑋(+g‘𝑈)𝑌) |
21 | fvex 6784 | . . . . 5 ⊢ (Base‘𝑆) ∈ V | |
22 | ressmpl.s | . . . . . . 7 ⊢ 𝑆 = (𝐼 mPoly 𝑅) | |
23 | eqid 2740 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
24 | 22, 9, 23 | mplval2 21200 | . . . . . 6 ⊢ 𝑆 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑆)) |
25 | eqid 2740 | . . . . . 6 ⊢ (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑅)) | |
26 | 24, 25 | ressplusg 16998 | . . . . 5 ⊢ ((Base‘𝑆) ∈ V → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘𝑆)) |
27 | 21, 26 | ax-mp 5 | . . . 4 ⊢ (+g‘(𝐼 mPwSer 𝑅)) = (+g‘𝑆) |
28 | fvex 6784 | . . . . 5 ⊢ (Base‘(𝐼 mPwSer 𝐻)) ∈ V | |
29 | 11, 25 | ressplusg 16998 | . . . . 5 ⊢ ((Base‘(𝐼 mPwSer 𝐻)) ∈ V → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))) |
30 | 28, 29 | ax-mp 5 | . . . 4 ⊢ (+g‘(𝐼 mPwSer 𝑅)) = (+g‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻)))) |
31 | ressmpl.p | . . . . . 6 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
32 | eqid 2740 | . . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
33 | 31, 32 | ressplusg 16998 | . . . . 5 ⊢ (𝐵 ∈ V → (+g‘𝑆) = (+g‘𝑃)) |
34 | 15, 33 | ax-mp 5 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑃) |
35 | 27, 30, 34 | 3eqtr3i 2776 | . . 3 ⊢ (+g‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻)))) = (+g‘𝑃) |
36 | 35 | oveqi 7284 | . 2 ⊢ (𝑋(+g‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))𝑌) = (𝑋(+g‘𝑃)𝑌) |
37 | 14, 20, 36 | 3eqtr3g 2803 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(+g‘𝑈)𝑌) = (𝑋(+g‘𝑃)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 ↾s cress 16939 +gcplusg 16960 SubRingcsubrg 20018 mPwSer cmps 21105 mPoly cmpl 21107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-tset 16979 df-subg 18750 df-ring 19783 df-subrg 20020 df-psr 21110 df-mpl 21112 |
This theorem is referenced by: ressply1add 21399 |
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