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| Mirrors > Home > MPE Home > Th. List > ringcval | Structured version Visualization version GIF version | ||
| Description: Value of the category of unital rings (in a universe). (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
| Ref | Expression |
|---|---|
| ringcval.c | ⊢ 𝐶 = (RingCat‘𝑈) |
| ringcval.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
| ringcval.b | ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
| ringcval.h | ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) |
| Ref | Expression |
|---|---|
| ringcval | ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ringcval.c | . 2 ⊢ 𝐶 = (RingCat‘𝑈) | |
| 2 | df-ringc 20646 | . . 3 ⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))))) | |
| 3 | fveq2 6906 | . . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈)) | |
| 4 | 3 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈)) |
| 5 | ineq1 4213 | . . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring)) | |
| 6 | 5 | sqxpeqd 5717 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) |
| 7 | ringcval.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) | |
| 8 | 7 | sqxpeqd 5717 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) |
| 9 | 8 | eqcomd 2743 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = (𝐵 × 𝐵)) |
| 10 | 6, 9 | sylan9eqr 2799 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = (𝐵 × 𝐵)) |
| 11 | 10 | reseq2d 5997 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = ( RingHom ↾ (𝐵 × 𝐵))) |
| 12 | ringcval.h | . . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
| 13 | 12 | eqcomd 2743 | . . . . . 6 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻) |
| 14 | 13 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻) |
| 15 | 11, 14 | eqtrd 2777 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = 𝐻) |
| 16 | 4, 15 | oveq12d 7449 | . . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
| 17 | ringcval.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
| 18 | 17 | elexd 3504 | . . 3 ⊢ (𝜑 → 𝑈 ∈ V) |
| 19 | ovexd 7466 | . . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V) | |
| 20 | 2, 16, 18, 19 | fvmptd2 7024 | . 2 ⊢ (𝜑 → (RingCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
| 21 | 1, 20 | eqtrid 2789 | 1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 × cxp 5683 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ↾cat cresc 17852 ExtStrCatcestrc 18166 Ringcrg 20230 RingHom crh 20469 RingCatcringc 20645 |
| 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-csb 3900 df-dif 3954 df-un 3956 df-in 3958 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-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-ringc 20646 |
| This theorem is referenced by: ringcbas 20650 ringchomfval 20651 ringccofval 20655 dfringc2 20657 ringccat 20663 ringcid 20664 funcringcsetc 20674 |
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