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Mirrors > Home > MPE Home > Th. List > psr1val | Structured version Visualization version GIF version |
Description: Value of the ring of univariate power series. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
psr1val.1 | ⊢ 𝑆 = (PwSer1‘𝑅) |
Ref | Expression |
---|---|
psr1val | ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psr1val.1 | . 2 ⊢ 𝑆 = (PwSer1‘𝑅) | |
2 | oveq2 7416 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅)) | |
3 | 2 | fveq1d 6893 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅)) |
4 | df-psr1 21703 | . . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅)) | |
5 | fvex 6904 | . . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V | |
6 | 3, 4, 5 | fvmpt 6998 | . . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
7 | 0fv 6935 | . . . . 5 ⊢ (∅‘∅) = ∅ | |
8 | 7 | eqcomi 2741 | . . . 4 ⊢ ∅ = (∅‘∅) |
9 | fvprc 6883 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
10 | reldmopsr 21599 | . . . . . 6 ⊢ Rel dom ordPwSer | |
11 | 10 | ovprc2 7448 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅) |
12 | 11 | fveq1d 6893 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅)) |
13 | 8, 9, 12 | 3eqtr4a 2798 | . . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
14 | 6, 13 | pm2.61i 182 | . 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅) |
15 | 1, 14 | eqtri 2760 | 1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4322 ‘cfv 6543 (class class class)co 7408 1oc1o 8458 ordPwSer copws 21460 PwSer1cps1 21698 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-opsr 21465 df-psr1 21703 |
This theorem is referenced by: psr1crng 21710 psr1assa 21711 psr1tos 21712 psr1bas2 21713 vr1cl2 21716 ply1lss 21719 ply1subrg 21720 psr1plusg 21743 psr1vsca 21744 psr1mulr 21745 psr1ring 21768 psr1lmod 21770 psr1sca 21771 |
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