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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 3515 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | elex 3515 | . . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
4 | id 22 | . . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
5 | fveq2 6673 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (ringLMod‘𝑟) = (ringLMod‘𝑅)) | |
6 | 5 | sneqd 4582 | . . . . . 6 ⊢ (𝑟 = 𝑅 → {(ringLMod‘𝑟)} = {(ringLMod‘𝑅)}) |
7 | 6 | xpeq2d 5588 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑖 × {(ringLMod‘𝑟)}) = (𝑖 × {(ringLMod‘𝑅)})) |
8 | 4, 7 | oveq12d 7177 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})) = (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)}))) |
9 | xpeq1 5572 | . . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 × {(ringLMod‘𝑅)}) = (𝐼 × {(ringLMod‘𝑅)})) | |
10 | 9 | oveq2d 7175 | . . . 4 ⊢ (𝑖 = 𝐼 → (𝑅 ⊕m (𝑖 × {(ringLMod‘𝑅)})) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
11 | df-frlm 20894 | . . . 4 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) | |
12 | ovex 7192 | . . . 4 ⊢ (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})) ∈ V | |
13 | 8, 10, 11, 12 | ovmpo 7313 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
14 | 2, 3, 13 | syl2an 597 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 freeLMod 𝐼) = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
15 | 1, 14 | syl5eq 2871 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 × cxp 5556 ‘cfv 6358 (class class class)co 7159 ringLModcrglmod 19944 ⊕m cdsmm 20878 freeLMod cfrlm 20893 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-frlm 20894 |
This theorem is referenced by: frlmlmod 20896 frlmpws 20897 frlmlss 20898 frlmpwsfi 20899 frlmbas 20902 |
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