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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 6113 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
2 | resspsr.2 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
3 | resspsr.h | . . . . . . . . 9 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
4 | 3 | subrgbas 18612 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻)) |
5 | 2, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑇 = (Base‘𝐻)) |
6 | eqid 2610 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 6 | subrgss 18604 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅)) |
8 | 2, 7 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅)) |
9 | 5, 8 | eqsstr3d 3603 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅)) |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝐻) ⊆ (Base‘𝑅)) |
11 | mapss 7786 | . . . . 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 694 | . . . 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 2610 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
15 | eqid 2610 | . . . . 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 19199 | . . . 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 2610 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
21 | 19, 6, 15, 20, 17 | psrbas 19199 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝑆) = ((Base‘𝑅) ↑𝑚 {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
22 | 12, 18, 21 | 3sstr4d 3611 | . . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆)) |
23 | reldmpsr 19182 | . . . . . . . . 9 ⊢ Rel dom mPwSer | |
24 | 23 | ovprc1 6582 | . . . . . . . 8 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPwSer 𝐻) = ∅) |
25 | 13, 24 | syl5eq 2656 | . . . . . . 7 ⊢ (¬ 𝐼 ∈ V → 𝑈 = ∅) |
26 | 25 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝑈 = ∅) |
27 | 26 | fveq2d 6107 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → (Base‘𝑈) = (Base‘∅)) |
28 | base0 15740 | . . . . 5 ⊢ ∅ = (Base‘∅) | |
29 | 27, 16, 28 | 3eqtr4g 2669 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 = ∅) |
30 | 0ss 3924 | . . . 4 ⊢ ∅ ⊆ (Base‘𝑆) | |
31 | 29, 30 | syl6eqss 3618 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆)) |
32 | 22, 31 | pm2.61dan 828 | . 2 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆)) |
33 | resspsr.p | . . 3 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
34 | 33, 20 | ressbas2 15758 | . 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 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ◡ccnv 5037 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Fincfn 7841 ℕcn 10897 ℕ0cn0 11169 Basecbs 15695 ↾s cress 15696 SubRingcsubrg 18599 mPwSer cmps 19172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-subg 17414 df-ring 18372 df-subrg 18601 df-psr 19177 |
This theorem is referenced by: resspsrvsca 19239 subrgpsr 19240 |
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