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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 7439 | . . . . 5 ⊢ (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅)) | |
| 3 | 2 | fveq1d 6908 | . . . 4 ⊢ (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅)) |
| 4 | df-vr1 22182 | . . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅)) | |
| 5 | fvex 6919 | . . . 4 ⊢ ((1o mVar 𝑅)‘∅) ∈ V | |
| 6 | 3, 4, 5 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1o mVar 𝑅)‘∅)) |
| 7 | 1, 6 | eqtrid 2789 | . 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅)) |
| 8 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅) | |
| 9 | 0fv 6950 | . . . 4 ⊢ (∅‘∅) = ∅ | |
| 10 | 8, 1, 9 | 3eqtr4g 2802 | . . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅)) |
| 11 | df-mvr 21930 | . . . . . 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 7567 | . . . . 5 ⊢ Rel dom mVar |
| 13 | 12 | ovprc2 7471 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mVar 𝑅) = ∅) |
| 14 | 13 | fveq1d 6908 | . . 3 ⊢ (¬ 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅)) |
| 15 | 10, 14 | eqtr4d 2780 | . 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅)) |
| 16 | 7, 15 | pm2.61i 182 | 1 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∅c0 4333 ifcif 4525 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 ↑m cmap 8866 Fincfn 8985 0cc0 11155 1c1 11156 ℕcn 12266 ℕ0cn0 12526 0gc0g 17484 1rcur 20178 mVar cmvr 21925 var1cv1 22177 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-mvr 21930 df-vr1 22182 |
| This theorem is referenced by: vr1cl2 22194 vr1cl 22219 subrgvr1 22264 subrgvr1cl 22265 coe1tm 22276 ply1coe 22302 evl1var 22340 evls1var 22342 rhmply1vr1 22391 |
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