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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 7364 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅)) | |
| 3 | 2 | fveq1d 6831 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅)) |
| 4 | df-psr1 22132 | . . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅)) | |
| 5 | fvex 6842 | . . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V | |
| 6 | 3, 4, 5 | fvmpt 6936 | . . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
| 7 | 0fv 6870 | . . . . 5 ⊢ (∅‘∅) = ∅ | |
| 8 | 7 | eqcomi 2744 | . . . 4 ⊢ ∅ = (∅‘∅) |
| 9 | fvprc 6821 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
| 10 | reldmopsr 22012 | . . . . . 6 ⊢ Rel dom ordPwSer | |
| 11 | 10 | ovprc2 7396 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅) |
| 12 | 11 | fveq1d 6831 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅)) |
| 13 | 8, 9, 12 | 3eqtr4a 2796 | . . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
| 14 | 6, 13 | pm2.61i 182 | . 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅) |
| 15 | 1, 14 | eqtri 2758 | 1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2114 Vcvv 3427 ∅c0 4263 ‘cfv 6487 (class class class)co 7356 1oc1o 8387 ordPwSer copws 21877 PwSer1cps1 22127 |
| 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 2184 ax-ext 2707 ax-sep 5220 ax-nul 5230 ax-pr 5364 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-ral 3050 df-rex 3060 df-rab 3388 df-v 3429 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4264 df-if 4457 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-br 5075 df-opab 5137 df-mpt 5156 df-id 5515 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-iota 6443 df-fun 6489 df-fv 6495 df-ov 7359 df-oprab 7360 df-mpo 7361 df-opsr 21882 df-psr1 22132 |
| This theorem is referenced by: psr1crng 22139 psr1assa 22140 psr1tos 22141 psr1bas2 22142 vr1cl2 22145 ply1lss 22148 ply1subrg 22149 psr1plusg 22172 psr1vsca 22173 psr1mulr 22174 psr1ring 22198 psr1lmod 22200 psr1sca 22201 |
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