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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 7153 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅)) | |
3 | 2 | fveq1d 6665 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅)) |
4 | df-vr1 20277 | . . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅)) | |
5 | fvex 6676 | . . . 4 ⊢ ((1o mVar 𝑅)‘∅) ∈ V | |
6 | 3, 4, 5 | fvmpt 6761 | . . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1o mVar 𝑅)‘∅)) |
7 | 1, 6 | syl5eq 2865 | . 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅)) |
8 | fvprc 6656 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅) | |
9 | 0fv 6702 | . . . 4 ⊢ (∅‘∅) = ∅ | |
10 | 8, 1, 9 | 3eqtr4g 2878 | . . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅)) |
11 | df-mvr 20065 | . . . . . 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 7274 | . . . . 5 ⊢ Rel dom mVar |
13 | 12 | ovprc2 7185 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mVar 𝑅) = ∅) |
14 | 13 | fveq1d 6665 | . . 3 ⊢ (¬ 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅)) |
15 | 10, 14 | eqtr4d 2856 | . 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅)) |
16 | 7, 15 | pm2.61i 183 | 1 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ∅c0 4288 ifcif 4463 ↦ cmpt 5137 ◡ccnv 5547 “ cima 5551 ‘cfv 6348 (class class class)co 7145 1oc1o 8084 ↑m cmap 8395 Fincfn 8497 0cc0 10525 1c1 10526 ℕcn 11626 ℕ0cn0 11885 0gc0g 16701 1rcur 19180 mVar cmvr 20060 var1cv1 20272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-mvr 20065 df-vr1 20277 |
This theorem is referenced by: vr1cl2 20289 vr1cl 20313 subrgvr1 20357 subrgvr1cl 20358 coe1tm 20369 ply1coe 20392 evl1var 20427 evls1var 20429 |
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