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Mirrors > Home > MPE Home > Th. List > rngcval | Structured version Visualization version GIF version |
Description: Value of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
Ref | Expression |
---|---|
rngcval.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rngcval.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcval.b | ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
rngcval.h | ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rngcval | ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcval.c | . 2 ⊢ 𝐶 = (RngCat‘𝑈) | |
2 | df-rngc 20555 | . . 3 ⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))) | |
3 | fveq2 6900 | . . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈)) | |
4 | 3 | adantl 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈)) |
5 | ineq1 4205 | . . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng)) | |
6 | 5 | sqxpeqd 5712 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) |
7 | rngcval.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
8 | 7 | sqxpeqd 5712 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) |
9 | 8 | eqcomd 2733 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵)) |
10 | 6, 9 | sylan9eqr 2789 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵)) |
11 | 10 | reseq2d 5987 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHom ↾ (𝐵 × 𝐵))) |
12 | rngcval.h | . . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) | |
13 | 12 | eqcomd 2733 | . . . . . 6 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) = 𝐻) |
14 | 13 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHom ↾ (𝐵 × 𝐵)) = 𝐻) |
15 | 11, 14 | eqtrd 2767 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻) |
16 | 4, 15 | oveq12d 7442 | . . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
17 | rngcval.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
18 | 17 | elexd 3492 | . . 3 ⊢ (𝜑 → 𝑈 ∈ V) |
19 | ovexd 7459 | . . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V) | |
20 | 2, 16, 18, 19 | fvmptd2 7016 | . 2 ⊢ (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
21 | 1, 20 | eqtrid 2779 | 1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ∩ cin 3946 × cxp 5678 ↾ cres 5682 ‘cfv 6551 (class class class)co 7424 ↾cat cresc 17796 ExtStrCatcestrc 18117 Rngcrng 20097 RngHom crnghm 20378 RngCatcrngc 20554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-res 5692 df-iota 6503 df-fun 6553 df-fv 6559 df-ov 7427 df-rngc 20555 |
This theorem is referenced by: rngcbas 20559 rngchomfval 20560 rngccofval 20564 dfrngc2 20566 rngccat 20572 rngcid 20573 rngcifuestrc 20577 funcrngcsetc 20578 |
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