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Mirrors > Home > MPE Home > Th. List > vr1val | Structured version Visualization version GIF version |
Description: The value of the generator of the power series algebra (the 𝑋 in 𝑅[[𝑋]]). Since all univariate polynomial rings over a fixed base ring 𝑅 are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and 1o = {∅} is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
vr1val.1 | ⊢ 𝑋 = (var1‘𝑅) |
Ref | Expression |
---|---|
vr1val | ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vr1val.1 | . . 3 ⊢ 𝑋 = (var1‘𝑅) | |
2 | oveq2 7158 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅)) | |
3 | 2 | fveq1d 6667 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅)) |
4 | df-vr1 20343 | . . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅)) | |
5 | fvex 6678 | . . . 4 ⊢ ((1o mVar 𝑅)‘∅) ∈ V | |
6 | 3, 4, 5 | fvmpt 6763 | . . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1o mVar 𝑅)‘∅)) |
7 | 1, 6 | syl5eq 2868 | . 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅)) |
8 | fvprc 6658 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅) | |
9 | 0fv 6704 | . . . 4 ⊢ (∅‘∅) = ∅ | |
10 | 8, 1, 9 | 3eqtr4g 2881 | . . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅)) |
11 | df-mvr 20131 | . . . . . 6 ⊢ mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑟), (0g‘𝑟))))) | |
12 | 11 | reldmmpo 7279 | . . . . 5 ⊢ Rel dom mVar |
13 | 12 | ovprc2 7190 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mVar 𝑅) = ∅) |
14 | 13 | fveq1d 6667 | . . 3 ⊢ (¬ 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅)) |
15 | 10, 14 | eqtr4d 2859 | . 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅)) |
16 | 7, 15 | pm2.61i 184 | 1 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3495 ∅c0 4291 ifcif 4467 ↦ cmpt 5139 ◡ccnv 5549 “ cima 5553 ‘cfv 6350 (class class class)co 7150 1oc1o 8089 ↑m cmap 8400 Fincfn 8503 0cc0 10531 1c1 10532 ℕcn 11632 ℕ0cn0 11891 0gc0g 16707 1rcur 19245 mVar cmvr 20126 var1cv1 20338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-mvr 20131 df-vr1 20343 |
This theorem is referenced by: vr1cl2 20355 vr1cl 20379 subrgvr1 20423 subrgvr1cl 20424 coe1tm 20435 ply1coe 20458 evl1var 20493 evls1var 20495 |
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