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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 7143 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅)) | |
3 | 2 | fveq1d 6647 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅)) |
4 | df-psr1 20809 | . . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅)) | |
5 | fvex 6658 | . . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V | |
6 | 3, 4, 5 | fvmpt 6745 | . . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
7 | 0fv 6684 | . . . . 5 ⊢ (∅‘∅) = ∅ | |
8 | 7 | eqcomi 2807 | . . . 4 ⊢ ∅ = (∅‘∅) |
9 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
10 | reldmopsr 20713 | . . . . . 6 ⊢ Rel dom ordPwSer | |
11 | 10 | ovprc2 7175 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅) |
12 | 11 | fveq1d 6647 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅)) |
13 | 8, 9, 12 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
14 | 6, 13 | pm2.61i 185 | . 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅) |
15 | 1, 14 | eqtri 2821 | 1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 ordPwSer copws 20593 PwSer1cps1 20804 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-opsr 20598 df-psr1 20809 |
This theorem is referenced by: psr1crng 20816 psr1assa 20817 psr1tos 20818 psr1bas2 20819 vr1cl2 20822 ply1lss 20825 ply1subrg 20826 psr1plusg 20851 psr1vsca 20852 psr1mulr 20853 psr1ring 20876 psr1lmod 20878 psr1sca 20879 |
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