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Mirrors > Home > MPE Home > Th. List > resspsrbas | Structured version Visualization version GIF version |
Description: A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.) |
Ref | Expression |
---|---|
resspsr.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
resspsr.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
resspsr.u | ⊢ 𝑈 = (𝐼 mPwSer 𝐻) |
resspsr.b | ⊢ 𝐵 = (Base‘𝑈) |
resspsr.p | ⊢ 𝑃 = (𝑆 ↾s 𝐵) |
resspsr.2 | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
Ref | Expression |
---|---|
resspsrbas | ⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6239 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
2 | resspsr.2 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
3 | resspsr.h | . . . . . . . . 9 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
4 | 3 | subrgbas 18837 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻)) |
5 | 2, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑇 = (Base‘𝐻)) |
6 | eqid 2651 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 6 | subrgss 18829 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅)) |
8 | 2, 7 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅)) |
9 | 5, 8 | eqsstr3d 3673 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅)) |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝐻) ⊆ (Base‘𝑅)) |
11 | mapss 7942 | . . . . 5 ⊢ (((Base‘𝑅) ∈ V ∧ (Base‘𝐻) ⊆ (Base‘𝑅)) → ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) | |
12 | 1, 10, 11 | sylancr 696 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ⊆ ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
13 | resspsr.u | . . . . 5 ⊢ 𝑈 = (𝐼 mPwSer 𝐻) | |
14 | eqid 2651 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
15 | eqid 2651 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
16 | resspsr.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑈) | |
17 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐼 ∈ V) | |
18 | 13, 14, 15, 16, 17 | psrbas 19426 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐵 = ((Base‘𝐻) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
19 | resspsr.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
20 | eqid 2651 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
21 | 19, 6, 15, 20, 17 | psrbas 19426 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝑆) = ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
22 | 12, 18, 21 | 3sstr4d 3681 | . . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆)) |
23 | reldmpsr 19409 | . . . . . . . . 9 ⊢ Rel dom mPwSer | |
24 | 23 | ovprc1 6724 | . . . . . . . 8 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPwSer 𝐻) = ∅) |
25 | 13, 24 | syl5eq 2697 | . . . . . . 7 ⊢ (¬ 𝐼 ∈ V → 𝑈 = ∅) |
26 | 25 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝑈 = ∅) |
27 | 26 | fveq2d 6233 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → (Base‘𝑈) = (Base‘∅)) |
28 | base0 15959 | . . . . 5 ⊢ ∅ = (Base‘∅) | |
29 | 27, 16, 28 | 3eqtr4g 2710 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 = ∅) |
30 | 0ss 4005 | . . . 4 ⊢ ∅ ⊆ (Base‘𝑆) | |
31 | 29, 30 | syl6eqss 3688 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆)) |
32 | 22, 31 | pm2.61dan 849 | . 2 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆)) |
33 | resspsr.p | . . 3 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
34 | 33, 20 | ressbas2 15978 | . 2 ⊢ (𝐵 ⊆ (Base‘𝑆) → 𝐵 = (Base‘𝑃)) |
35 | 32, 34 | syl 17 | 1 ⊢ (𝜑 → 𝐵 = (Base‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 {crab 2945 Vcvv 3231 ⊆ wss 3607 ∅c0 3948 ◡ccnv 5142 “ cima 5146 ‘cfv 5926 (class class class)co 6690 ↑𝑚 cmap 7899 Fincfn 7997 ℕcn 11058 ℕ0cn0 11330 Basecbs 15904 ↾s cress 15905 SubRingcsubrg 18824 mPwSer cmps 19399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-tset 16007 df-subg 17638 df-ring 18595 df-subrg 18826 df-psr 19404 |
This theorem is referenced by: resspsrvsca 19466 subrgpsr 19467 |
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