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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 6909 | . . . . 5 ⊢ (Base‘𝑅) ∈ V | |
2 | resspsr.2 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
3 | resspsr.h | . . . . . . . . 9 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
4 | 3 | subrgbas 20532 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻)) |
5 | 2, 4 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑇 = (Base‘𝐻)) |
6 | eqid 2725 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 6 | subrgss 20523 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅)) |
8 | 2, 7 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅)) |
9 | 5, 8 | eqsstrrd 4016 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅)) |
10 | 9 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝐻) ⊆ (Base‘𝑅)) |
11 | mapss 8908 | . . . . 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 585 | . . . 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 2725 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
15 | eqid 2725 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
16 | resspsr.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑈) | |
17 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐼 ∈ V) | |
18 | 13, 14, 15, 16, 17 | psrbas 21895 | . . . 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 2725 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
21 | 19, 6, 15, 20, 17 | psrbas 21895 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝑆) = ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
22 | 12, 18, 21 | 3sstr4d 4024 | . . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆)) |
23 | reldmpsr 21864 | . . . . . . . . 9 ⊢ Rel dom mPwSer | |
24 | 23 | ovprc1 7458 | . . . . . . . 8 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPwSer 𝐻) = ∅) |
25 | 13, 24 | eqtrid 2777 | . . . . . . 7 ⊢ (¬ 𝐼 ∈ V → 𝑈 = ∅) |
26 | 25 | adantl 480 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝑈 = ∅) |
27 | 26 | fveq2d 6900 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → (Base‘𝑈) = (Base‘∅)) |
28 | base0 17188 | . . . . 5 ⊢ ∅ = (Base‘∅) | |
29 | 27, 16, 28 | 3eqtr4g 2790 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 = ∅) |
30 | 0ss 4398 | . . . 4 ⊢ ∅ ⊆ (Base‘𝑆) | |
31 | 29, 30 | eqsstrdi 4031 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆)) |
32 | 22, 31 | pm2.61dan 811 | . 2 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆)) |
33 | resspsr.p | . . 3 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
34 | 33, 20 | ressbas2 17221 | . 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 394 = wceq 1533 ∈ wcel 2098 {crab 3418 Vcvv 3461 ⊆ wss 3944 ∅c0 4322 ◡ccnv 5677 “ cima 5681 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 Fincfn 8964 ℕcn 12245 ℕ0cn0 12505 Basecbs 17183 ↾s cress 17212 SubRingcsubrg 20518 mPwSer cmps 21854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-tset 17255 df-subg 19086 df-ring 20187 df-subrg 20520 df-psr 21859 |
This theorem is referenced by: resspsrvsca 21939 subrgpsr 21940 |
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