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Theorem psrbaspropd 22198

Description: Property deduction for power series base set. (Contributed by Stefan O'Rear, 27-Mar-2015.)

**Hypothesis**
Ref	Expression
psrbaspropd.e	⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑆))

**Assertion**
Ref	Expression
psrbaspropd	⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))

Proof of Theorem psrbaspropd Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
2		eqid 2736	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2736	. . . 4 ⊢ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}
4		eqid 2736	. . . 4 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
5		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → 𝐼 ∈ V)
6	1, 2, 3, 4, 5	psrbas 21913	. . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑_m {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}))
7		eqid 2736	. . . . 5 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆)
8		eqid 2736	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2736	. . . . 5 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆))
10	7, 8, 3, 9, 5	psrbas 21913	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘(𝐼 mPwSer 𝑆)) = ((Base‘𝑆) ↑_m {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}))
11		psrbaspropd.e	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑆))
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘𝑅) = (Base‘𝑆))
13	12	oveq1d 7382	. . . 4 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → ((Base‘𝑅) ↑_m {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) = ((Base‘𝑆) ↑_m {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}))
14	10, 13	eqtr4d 2774	. . 3 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘(𝐼 mPwSer 𝑆)) = ((Base‘𝑅) ↑_m {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}))
15	6, 14	eqtr4d 2774	. 2 ⊢ ((𝜑 ∧ 𝐼 ∈ V) → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
16		reldmpsr 21894	. . . . . 6 ⊢ Rel dom mPwSer
17	16	ovprc1 7406	. . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPwSer 𝑅) = ∅)
18	16	ovprc1 7406	. . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPwSer 𝑆) = ∅)
19	17, 18	eqtr4d 2774	. . . 4 ⊢ (¬ 𝐼 ∈ V → (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑆))
20	19	fveq2d 6844	. . 3 ⊢ (¬ 𝐼 ∈ V → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
21	20	adantl 481	. 2 ⊢ ((𝜑 ∧ ¬ 𝐼 ∈ V) → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
22	15, 21	pm2.61dan 813	1 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3389 Vcvv 3429 ∅c0 4273 ^◡ccnv 5630 “ cima 5634 ‘cfv 6498 (class class class)co 7367 ↑_m cmap 8773 Fincfn 8893 ℕcn 12174 ℕ₀cn0 12437 Basecbs 17179 mPwSer cmps 21884

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-tset 17239 df-psr 21889

This theorem is referenced by: psrplusgpropd 22199 mplbaspropd 22200 psropprmul 22201

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