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Theorem resspsrbas 21965

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 6848	. . . . 5 ⊢ (Base‘𝑅) ∈ V
2		resspsr.2	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))
3		resspsr.h	. . . . . . . . 9 ⊢ 𝐻 = (𝑅 ↾_s 𝑇)
4	3	subrgbas 20552	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻))
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 = (Base‘𝐻))
6		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
7	6	subrgss 20543	. . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅))
8	2, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅))
9	5, 8	eqsstrrd 3958	. . . . . 6 ⊢ (𝜑 → (Base‘𝐻) ⊆ (Base‘𝑅))
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝐻) ⊆ (Base‘𝑅))
11		mapss 8831	. . . . 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 588	. . . 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 2737	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
15		eqid 2737	. . . . 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 21926	. . . 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 2737	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
21	19, 6, 15, 20, 17	psrbas 21926	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝑆) = ((Base‘𝑅) ↑_m {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}))
22	12, 18, 21	3sstr4d 3978	. . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐵 ⊆ (Base‘𝑆))
23		reldmpsr 21907	. . . . . . . . 9 ⊢ Rel dom mPwSer
24	23	ovprc1 7400	. . . . . . . 8 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPwSer 𝐻) = ∅)
25	13, 24	eqtrid 2784	. . . . . . 7 ⊢ (¬ 𝐼 ∈ V → 𝑈 = ∅)
26	25	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝑈 = ∅)
27	26	fveq2d 6839	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → (Base‘𝑈) = (Base‘∅))
28		base0 17178	. . . . 5 ⊢ ∅ = (Base‘∅)
29	27, 16, 28	3eqtr4g 2797	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → 𝐵 = ∅)
30		0ss 4341	. . . 4 ⊢ ∅ ⊆ (Base‘𝑆)
31	29, 30	eqsstrdi 3967	. . 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 17202	. 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 1542 ∈ wcel 2114 {crab 3390 Vcvv 3430 ⊆ wss 3890 ∅c0 4274 ^◡ccnv 5624 “ cima 5628 ‘cfv 6493 (class class class)co 7361 ↑_m cmap 8767 Fincfn 8887 ℕcn 12168 ℕ₀cn0 12431 Basecbs 17173 ↾_s cress 17194 SubRingcsubrg 20540 mPwSer cmps 21897

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-tset 17233 df-subg 19093 df-ring 20210 df-subrg 20541 df-psr 21902

This theorem is referenced by: resspsrvsca 21968 subrgpsr 21969

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