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Theorem resspsrbas 21911

Description: A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
resspsrbas	⊢ (𝜑 → 𝐵 = (Base‘𝑃))

Proof of Theorem resspsrbas Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6835	. . . . 5 ⊢ (Base‘𝑅) ∈ V
2		resspsr.2	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))
3		resspsr.h	. . . . . . . . 9 ⊢ 𝐻 = (𝑅 ↾_s 𝑇)
4	3	subrgbas 20496	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 = (Base‘𝐻))
6		eqid 2731	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
7	6	subrgss 20487	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
8	2, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅))
9	5, 8	eqsstrrd 3965	. . . . . 6 ⊢ (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅))
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝐻) ⊆ (Base‘𝑅))
11		mapss 8813	. . . . 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 587	. . . 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 2731	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
15		eqid 2731	. . . . 5 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
16		resspsr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑈)
17		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐼 ∈ V)
18	13, 14, 15, 16, 17	psrbas 21870	. . . 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 2731	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	19, 6, 15, 20, 17	psrbas 21870	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝑆) = ((Base‘𝑅) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
22	12, 18, 21	3sstr4d 3985	. . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆))
23		reldmpsr 21851	. . . . . . . . 9 ⊢ Rel dom mPwSer
24	23	ovprc1 7385	. . . . . . . 8 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPwSer 𝐻) = ∅)
25	13, 24	eqtrid 2778	. . . . . . 7 ⊢ (¬ 𝐼 ∈ V → 𝑈 = ∅)
26	25	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝑈 = ∅)
27	26	fveq2d 6826	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → (Base‘𝑈) = (Base‘∅))
28		base0 17125	. . . . 5 ⊢ ∅ = (Base‘∅)
29	27, 16, 28	3eqtr4g 2791	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 = ∅)
30		0ss 4347	. . . 4 ⊢ ∅ ⊆ (Base‘𝑆)
31	29, 30	eqsstrdi 3974	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆))
32	22, 31	pm2.61dan 812	. 2 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆))
33		resspsr.p	. . 3 ⊢ 𝑃 = (𝑆 ↾_s 𝐵)
34	33, 20	ressbas2 17149	. 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 395 = wceq 1541 ∈ wcel 2111 {crab 3395 Vcvv 3436 ⊆ wss 3897 ∅c0 4280 ^◡ccnv 5613 “ cima 5617 ‘cfv 6481 (class class class)co 7346 ↑_m cmap 8750 Fincfn 8869 ℕcn 12125 ℕ₀cn0 12381 Basecbs 17120 ↾_s cress 17141 SubRingcsubrg 20484 mPwSer cmps 21841

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-tset 17180 df-subg 19036 df-ring 20153 df-subrg 20485 df-psr 21846

This theorem is referenced by: resspsrvsca 21914 subrgpsr 21915

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