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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 6914 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅)) | |
3 | 2 | fveq1d 6436 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅)) |
4 | df-psr1 19911 | . . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅)) | |
5 | fvex 6447 | . . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V | |
6 | 3, 4, 5 | fvmpt 6530 | . . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
7 | 0fv 6474 | . . . . 5 ⊢ (∅‘∅) = ∅ | |
8 | 7 | eqcomi 2835 | . . . 4 ⊢ ∅ = (∅‘∅) |
9 | fvprc 6427 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
10 | reldmopsr 19835 | . . . . . 6 ⊢ Rel dom ordPwSer | |
11 | 10 | ovprc2 6945 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅) |
12 | 11 | fveq1d 6436 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅)) |
13 | 8, 9, 12 | 3eqtr4a 2888 | . . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
14 | 6, 13 | pm2.61i 177 | . 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅) |
15 | 1, 14 | eqtri 2850 | 1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1658 ∈ wcel 2166 Vcvv 3415 ∅c0 4145 ‘cfv 6124 (class class class)co 6906 1oc1o 7820 ordPwSer copws 19717 PwSer1cps1 19906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-iota 6087 df-fun 6126 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-opsr 19722 df-psr1 19911 |
This theorem is referenced by: psr1crng 19918 psr1assa 19919 psr1tos 19920 psr1bas2 19921 vr1cl2 19924 ply1lss 19927 ply1subrg 19928 psr1plusg 19953 psr1vsca 19954 psr1mulr 19955 psr1ring 19978 psr1lmod 19980 psr1sca 19981 |
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