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Mirrors > Home > MPE Home > Th. List > frlmrcl | Structured version Visualization version GIF version |
Description: If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
Ref | Expression |
---|---|
frlmval.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
frlmrcl.b | ⊢ 𝐵 = (Base‘𝐹) |
Ref | Expression |
---|---|
frlmrcl | ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmval.f | . 2 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
2 | frlmrcl.b | . 2 ⊢ 𝐵 = (Base‘𝐹) | |
3 | df-frlm 20593 | . . 3 ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) | |
4 | 3 | reldmmpo 7101 | . 2 ⊢ Rel dom freeLMod |
5 | 1, 2, 4 | strov2rcl 16402 | 1 ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 Vcvv 3416 {csn 4441 × cxp 5405 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 ringLModcrglmod 19663 ⊕m cdsmm 20577 freeLMod cfrlm 20592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-iota 6152 df-fun 6190 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-slot 16343 df-base 16345 df-frlm 20593 |
This theorem is referenced by: frlmbasfsupp 20604 frlmbasmap 20605 frlmvscafval 20612 |
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