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Theorem resspsrbas 22011

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 6919	. . . . 5 ⊢ (Base‘𝑅) ∈ V
2		resspsr.2	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))
3		resspsr.h	. . . . . . . . 9 ⊢ 𝐻 = (𝑅 ↾_s 𝑇)
4	3	subrgbas 20597	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 = (Base‘𝐻))
6		eqid 2734	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
7	6	subrgss 20588	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
8	2, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅))
9	5, 8	eqsstrrd 4034	. . . . . 6 ⊢ (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅))
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝐻) ⊆ (Base‘𝑅))
11		mapss 8927	. . . . 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 2734	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
15		eqid 2734	. . . . 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 21970	. . . 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 2734	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	19, 6, 15, 20, 17	psrbas 21970	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝑆) = ((Base‘𝑅) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
22	12, 18, 21	3sstr4d 4042	. . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆))
23		reldmpsr 21951	. . . . . . . . 9 ⊢ Rel dom mPwSer
24	23	ovprc1 7469	. . . . . . . 8 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPwSer 𝐻) = ∅)
25	13, 24	eqtrid 2786	. . . . . . 7 ⊢ (¬ 𝐼 ∈ V → 𝑈 = ∅)
26	25	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝑈 = ∅)
27	26	fveq2d 6910	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → (Base‘𝑈) = (Base‘∅))
28		base0 17249	. . . . 5 ⊢ ∅ = (Base‘∅)
29	27, 16, 28	3eqtr4g 2799	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 = ∅)
30		0ss 4405	. . . 4 ⊢ ∅ ⊆ (Base‘𝑆)
31	29, 30	eqsstrdi 4049	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆))
32	22, 31	pm2.61dan 813	. 2 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆))
33		resspsr.p	. . 3 ⊢ 𝑃 = (𝑆 ↾_s 𝐵)
34	33, 20	ressbas2 17282	. 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 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 ⊆ wss 3962 ∅c0 4338 ^◡ccnv 5687 “ cima 5691 ‘cfv 6562 (class class class)co 7430 ↑_m cmap 8864 Fincfn 8983 ℕcn 12263 ℕ₀cn0 12523 Basecbs 17244 ↾_s cress 17273 SubRingcsubrg 20585 mPwSer cmps 21941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-subg 19153 df-ring 20252 df-subrg 20586 df-psr 21946

This theorem is referenced by: resspsrvsca 22014 subrgpsr 22015

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