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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 7143 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅)) | |
3 | 2 | fveq1d 6647 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅)) |
4 | df-vr1 20810 | . . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅)) | |
5 | fvex 6658 | . . . 4 ⊢ ((1o mVar 𝑅)‘∅) ∈ V | |
6 | 3, 4, 5 | fvmpt 6745 | . . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1o mVar 𝑅)‘∅)) |
7 | 1, 6 | syl5eq 2845 | . 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅)) |
8 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅) | |
9 | 0fv 6684 | . . . 4 ⊢ (∅‘∅) = ∅ | |
10 | 8, 1, 9 | 3eqtr4g 2858 | . . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅)) |
11 | df-mvr 20595 | . . . . . 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 7264 | . . . . 5 ⊢ Rel dom mVar |
13 | 12 | ovprc2 7175 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mVar 𝑅) = ∅) |
14 | 13 | fveq1d 6647 | . . 3 ⊢ (¬ 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅)) |
15 | 10, 14 | eqtr4d 2836 | . 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅)) |
16 | 7, 15 | pm2.61i 185 | 1 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ∅c0 4243 ifcif 4425 ↦ cmpt 5110 ◡ccnv 5518 “ cima 5522 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 ↑m cmap 8389 Fincfn 8492 0cc0 10526 1c1 10527 ℕcn 11625 ℕ0cn0 11885 0gc0g 16705 1rcur 19244 mVar cmvr 20590 var1cv1 20805 |
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-mvr 20595 df-vr1 20810 |
This theorem is referenced by: vr1cl2 20822 vr1cl 20846 subrgvr1 20890 subrgvr1cl 20891 coe1tm 20902 ply1coe 20925 evl1var 20960 evls1var 20962 |
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