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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 7166 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o ordPwSer 𝑟) = (1o ordPwSer 𝑅)) | |
3 | 2 | fveq1d 6674 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o ordPwSer 𝑟)‘∅) = ((1o ordPwSer 𝑅)‘∅)) |
4 | df-psr1 20350 | . . . 4 ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1o ordPwSer 𝑟)‘∅)) | |
5 | fvex 6685 | . . . 4 ⊢ ((1o ordPwSer 𝑅)‘∅) ∈ V | |
6 | 3, 4, 5 | fvmpt 6770 | . . 3 ⊢ (𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
7 | 0fv 6711 | . . . . 5 ⊢ (∅‘∅) = ∅ | |
8 | 7 | eqcomi 2832 | . . . 4 ⊢ ∅ = (∅‘∅) |
9 | fvprc 6665 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ∅) | |
10 | reldmopsr 20256 | . . . . . 6 ⊢ Rel dom ordPwSer | |
11 | 10 | ovprc2 7198 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o ordPwSer 𝑅) = ∅) |
12 | 11 | fveq1d 6674 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ((1o ordPwSer 𝑅)‘∅) = (∅‘∅)) |
13 | 8, 9, 12 | 3eqtr4a 2884 | . . 3 ⊢ (¬ 𝑅 ∈ V → (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅)) |
14 | 6, 13 | pm2.61i 184 | . 2 ⊢ (PwSer1‘𝑅) = ((1o ordPwSer 𝑅)‘∅) |
15 | 1, 14 | eqtri 2846 | 1 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 ‘cfv 6357 (class class class)co 7158 1oc1o 8097 ordPwSer copws 20137 PwSer1cps1 20345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-opsr 20142 df-psr1 20350 |
This theorem is referenced by: psr1crng 20357 psr1assa 20358 psr1tos 20359 psr1bas2 20360 vr1cl2 20363 ply1lss 20366 ply1subrg 20367 psr1plusg 20392 psr1vsca 20393 psr1mulr 20394 psr1ring 20417 psr1lmod 20419 psr1sca 20420 |
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