|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 3501 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
| 3 | elex 3501 | . . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
| 4 | id 22 | . . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
| 5 | fveq2 6906 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (ringLMod‘𝑟) = (ringLMod‘𝑅)) | |
| 6 | 5 | sneqd 4638 | . . . . . 6 ⊢ (𝑟 = 𝑅 → {(ringLMod‘𝑟)} = {(ringLMod‘𝑅)}) | 
| 7 | 6 | xpeq2d 5715 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑖 × {(ringLMod‘𝑟)}) = (𝑖 × {(ringLMod‘𝑅)})) | 
| 8 | 4, 7 | oveq12d 7449 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})) = (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)}))) | 
| 9 | xpeq1 5699 | . . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 × {(ringLMod‘𝑅)}) = (𝐼 × {(ringLMod‘𝑅)})) | |
| 10 | 9 | oveq2d 7447 | . . . 4 ⊢ (𝑖 = 𝐼 → (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)})) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) | 
| 11 | df-frlm 21767 | . . . 4 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) | |
| 12 | ovex 7464 | . . . 4 ⊢ (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) ∈ V | |
| 13 | 8, 10, 11, 12 | ovmpo 7593 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) | 
| 14 | 2, 3, 13 | syl2an 596 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) | 
| 15 | 1, 14 | eqtrid 2789 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ringLModcrglmod 21171 ⊕m cdsmm 21751 freeLMod cfrlm 21766 | 
| 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-sbc 3789 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-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-frlm 21767 | 
| This theorem is referenced by: frlmlmod 21769 frlmpws 21770 frlmlss 21771 frlmpwsfi 21772 frlmbas 21775 | 
| Copyright terms: Public domain | W3C validator |