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Theorem resspsrbas 21184

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 6787	. . . . 5 ⊢ (Base‘𝑅) ∈ V
2		resspsr.2	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))
3		resspsr.h	. . . . . . . . 9 ⊢ 𝐻 = (𝑅 ↾_s 𝑇)
4	3	subrgbas 20033	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 = (Base‘𝐻))
6		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
7	6	subrgss 20025	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
8	2, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅))
9	5, 8	eqsstrrd 3960	. . . . . 6 ⊢ (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅))
10	9	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝐻) ⊆ (Base‘𝑅))
11		mapss 8677	. . . . 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 2738	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
15		eqid 2738	. . . . 5 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
16		resspsr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑈)
17		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐼 ∈ V)
18	13, 14, 15, 16, 17	psrbas 21147	. . . 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 2738	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	19, 6, 15, 20, 17	psrbas 21147	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝑆) = ((Base‘𝑅) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
22	12, 18, 21	3sstr4d 3968	. . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆))
23		reldmpsr 21117	. . . . . . . . 9 ⊢ Rel dom mPwSer
24	23	ovprc1 7314	. . . . . . . 8 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPwSer 𝐻) = ∅)
25	13, 24	eqtrid 2790	. . . . . . 7 ⊢ (¬ 𝐼 ∈ V → 𝑈 = ∅)
26	25	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝑈 = ∅)
27	26	fveq2d 6778	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → (Base‘𝑈) = (Base‘∅))
28		base0 16917	. . . . 5 ⊢ ∅ = (Base‘∅)
29	27, 16, 28	3eqtr4g 2803	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 = ∅)
30		0ss 4330	. . . 4 ⊢ ∅ ⊆ (Base‘𝑆)
31	29, 30	eqsstrdi 3975	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆))
32	22, 31	pm2.61dan 810	. 2 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆))
33		resspsr.p	. . 3 ⊢ 𝑃 = (𝑆 ↾_s 𝐵)
34	33, 20	ressbas2 16949	. 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 396 = wceq 1539 ∈ wcel 2106 {crab 3068 Vcvv 3432 ⊆ wss 3887 ∅c0 4256 ^◡ccnv 5588 “ cima 5592 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 Fincfn 8733 ℕcn 11973 ℕ₀cn0 12233 Basecbs 16912 ↾_s cress 16941 SubRingcsubrg 20020 mPwSer cmps 21107

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-tset 16981 df-subg 18752 df-ring 19785 df-subrg 20022 df-psr 21112

This theorem is referenced by: resspsrvsca 21187 subrgpsr 21188

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