Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 = ( RngHomo ↾ (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 41751 | . . . 4 ⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))) |
4 | fveq2 6103 | . . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈)) | |
5 | 4 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈)) |
6 | ineq1 3769 | . . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng)) | |
7 | 6 | sqxpeqd 5065 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) |
8 | rngcval.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
9 | 8 | sqxpeqd 5065 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) |
10 | 9 | eqcomd 2616 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵)) |
11 | 7, 10 | sylan9eqr 2666 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵)) |
12 | 11 | reseq2d 5317 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHomo ↾ (𝐵 × 𝐵))) |
13 | rngcval.h | . . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
14 | 13 | eqcomd 2616 | . . . . . 6 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻) |
15 | 14 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻) |
16 | 12, 15 | eqtrd 2644 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻) |
17 | 5, 16 | oveq12d 6567 | . . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
18 | rngcval.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
19 | elex 3185 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ∈ V) |
21 | ovex 6577 | . . . 4 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V | |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V) |
23 | 3, 17, 20, 22 | fvmptd 6197 | . 2 ⊢ (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
24 | 1, 23 | syl5eq 2656 | 1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ↾cat cresc 16291 ExtStrCatcestrc 16585 Rngcrng 41664 RngHomo crngh 41675 RngCatcrngc 41749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-rngc 41751 |
This theorem is referenced by: rngcbas 41757 rngchomfval 41758 rngccofval 41762 dfrngc2 41764 rngccat 41770 rngcid 41771 rngcifuestrc 41789 funcrngcsetc 41790 |
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