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Mirrors > Home > MPE Home > Th. List > frlmval | Structured version Visualization version GIF version |
Description: Value of the "free module" function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
frlmval.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
Ref | Expression |
---|---|
frlmval | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmval.f | . 2 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
2 | elex 3439 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | elex 3439 | . . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
4 | id 22 | . . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
5 | fveq2 6736 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (ringLMod‘𝑟) = (ringLMod‘𝑅)) | |
6 | 5 | sneqd 4568 | . . . . . 6 ⊢ (𝑟 = 𝑅 → {(ringLMod‘𝑟)} = {(ringLMod‘𝑅)}) |
7 | 6 | xpeq2d 5596 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑖 × {(ringLMod‘𝑟)}) = (𝑖 × {(ringLMod‘𝑅)})) |
8 | 4, 7 | oveq12d 7250 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})) = (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)}))) |
9 | xpeq1 5580 | . . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 × {(ringLMod‘𝑅)}) = (𝐼 × {(ringLMod‘𝑅)})) | |
10 | 9 | oveq2d 7248 | . . . 4 ⊢ (𝑖 = 𝐼 → (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)})) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
11 | df-frlm 20734 | . . . 4 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) | |
12 | ovex 7265 | . . . 4 ⊢ (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) ∈ V | |
13 | 8, 10, 11, 12 | ovmpo 7388 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
14 | 2, 3, 13 | syl2an 599 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
15 | 1, 14 | eqtrid 2790 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 Vcvv 3421 {csn 4556 × cxp 5564 ‘cfv 6398 (class class class)co 7232 ringLModcrglmod 20231 ⊕m cdsmm 20718 freeLMod cfrlm 20733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-sbc 3710 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-iota 6356 df-fun 6400 df-fv 6406 df-ov 7235 df-oprab 7236 df-mpo 7237 df-frlm 20734 |
This theorem is referenced by: frlmlmod 20736 frlmpws 20737 frlmlss 20738 frlmpwsfi 20739 frlmbas 20742 |
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