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Mirrors > Home > MPE Home > Th. List > ressply1vsca | Structured version Visualization version GIF version |
Description: A restricted power series algebra has the same scalar 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 |
---|---|
ressply1vsca | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2772 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | ressply1.h | . . 3 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
3 | eqid 2772 | . . 3 ⊢ (1o mPoly 𝐻) = (1o mPoly 𝐻) | |
4 | ressply1.u | . . . 4 ⊢ 𝑈 = (Poly1‘𝐻) | |
5 | eqid 2772 | . . . 4 ⊢ (PwSer1‘𝐻) = (PwSer1‘𝐻) | |
6 | ressply1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑈) | |
7 | 4, 5, 6 | ply1bas 20056 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝐻)) |
8 | 1on 7904 | . . . 4 ⊢ 1o ∈ On | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ On) |
10 | ressply1.2 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
11 | eqid 2772 | . . 3 ⊢ ((1o mPoly 𝑅) ↾s 𝐵) = ((1o mPoly 𝑅) ↾s 𝐵) | |
12 | 1, 2, 3, 7, 9, 10, 11 | ressmplvsca 19943 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘(1o mPoly 𝐻))𝑌) = (𝑋( ·𝑠 ‘((1o mPoly 𝑅) ↾s 𝐵))𝑌)) |
13 | eqid 2772 | . . . 4 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
14 | 4, 3, 13 | ply1vsca 20087 | . . 3 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘(1o mPoly 𝐻)) |
15 | 14 | oveqi 6983 | . 2 ⊢ (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘(1o mPoly 𝐻))𝑌) |
16 | ressply1.s | . . . . 5 ⊢ 𝑆 = (Poly1‘𝑅) | |
17 | eqid 2772 | . . . . 5 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
18 | 16, 1, 17 | ply1vsca 20087 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘(1o mPoly 𝑅)) |
19 | 6 | fvexi 6507 | . . . . 5 ⊢ 𝐵 ∈ V |
20 | ressply1.p | . . . . . 6 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
21 | 20, 17 | ressvsca 16497 | . . . . 5 ⊢ (𝐵 ∈ V → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑃)) |
22 | 19, 21 | ax-mp 5 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑃) |
23 | eqid 2772 | . . . . . 6 ⊢ ( ·𝑠 ‘(1o mPoly 𝑅)) = ( ·𝑠 ‘(1o mPoly 𝑅)) | |
24 | 11, 23 | ressvsca 16497 | . . . . 5 ⊢ (𝐵 ∈ V → ( ·𝑠 ‘(1o mPoly 𝑅)) = ( ·𝑠 ‘((1o mPoly 𝑅) ↾s 𝐵))) |
25 | 19, 24 | ax-mp 5 | . . . 4 ⊢ ( ·𝑠 ‘(1o mPoly 𝑅)) = ( ·𝑠 ‘((1o mPoly 𝑅) ↾s 𝐵)) |
26 | 18, 22, 25 | 3eqtr3i 2804 | . . 3 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘((1o mPoly 𝑅) ↾s 𝐵)) |
27 | 26 | oveqi 6983 | . 2 ⊢ (𝑋( ·𝑠 ‘𝑃)𝑌) = (𝑋( ·𝑠 ‘((1o mPoly 𝑅) ↾s 𝐵))𝑌) |
28 | 12, 15, 27 | 3eqtr4g 2833 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 Vcvv 3409 Oncon0 6023 ‘cfv 6182 (class class class)co 6970 1oc1o 7890 Basecbs 16329 ↾s cress 16330 ·𝑠 cvsca 16415 SubRingcsubrg 19244 mPoly cmpl 19837 PwSer1cps1 20036 Poly1cpl1 20038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-sca 16427 df-vsca 16428 df-tset 16430 df-ple 16431 df-subg 18050 df-ring 19012 df-subrg 19246 df-psr 19840 df-mpl 19842 df-opsr 19844 df-psr1 20041 df-ply1 20043 |
This theorem is referenced by: (None) |
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