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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 7439 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅)) | |
| 3 | 2 | fveq1d 6908 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅)) |
| 4 | df-psr1 22181 | . . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅)) | |
| 5 | fvex 6919 | . . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V | |
| 6 | 3, 4, 5 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
| 7 | 0fv 6950 | . . . . 5 ⊢ (∅‘∅) = ∅ | |
| 8 | 7 | eqcomi 2746 | . . . 4 ⊢ ∅ = (∅‘∅) |
| 9 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
| 10 | reldmopsr 22063 | . . . . . 6 ⊢ Rel dom ordPwSer | |
| 11 | 10 | ovprc2 7471 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅) |
| 12 | 11 | fveq1d 6908 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅)) |
| 13 | 8, 9, 12 | 3eqtr4a 2803 | . . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
| 14 | 6, 13 | pm2.61i 182 | . 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅) |
| 15 | 1, 14 | eqtri 2765 | 1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 ordPwSer copws 21928 PwSer1cps1 22176 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-opsr 21933 df-psr1 22181 |
| This theorem is referenced by: psr1crng 22188 psr1assa 22189 psr1tos 22190 psr1bas2 22191 vr1cl2 22194 ply1lss 22198 ply1subrg 22199 psr1plusg 22222 psr1vsca 22223 psr1mulr 22224 psr1ring 22248 psr1lmod 22250 psr1sca 22251 |
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