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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 7428 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅)) | |
3 | 2 | fveq1d 6899 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅)) |
4 | df-psr1 22099 | . . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅)) | |
5 | fvex 6910 | . . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V | |
6 | 3, 4, 5 | fvmpt 7005 | . . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
7 | 0fv 6941 | . . . . 5 ⊢ (∅‘∅) = ∅ | |
8 | 7 | eqcomi 2737 | . . . 4 ⊢ ∅ = (∅‘∅) |
9 | fvprc 6889 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
10 | reldmopsr 21983 | . . . . . 6 ⊢ Rel dom ordPwSer | |
11 | 10 | ovprc2 7460 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅) |
12 | 11 | fveq1d 6899 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅)) |
13 | 8, 9, 12 | 3eqtr4a 2794 | . . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
14 | 6, 13 | pm2.61i 182 | . 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅) |
15 | 1, 14 | eqtri 2756 | 1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∅c0 4323 ‘cfv 6548 (class class class)co 7420 1oc1o 8480 ordPwSer copws 21841 PwSer1cps1 22094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-opsr 21846 df-psr1 22099 |
This theorem is referenced by: psr1crng 22106 psr1assa 22107 psr1tos 22108 psr1bas2 22109 vr1cl2 22112 ply1lss 22115 ply1subrg 22116 psr1plusg 22139 psr1vsca 22140 psr1mulr 22141 psr1ring 22165 psr1lmod 22167 psr1sca 22168 |
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