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 44250 | . . 3 ⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))) | |
3 | fveq2 6670 | . . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈)) | |
4 | 3 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈)) |
5 | ineq1 4181 | . . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng)) | |
6 | 5 | sqxpeqd 5587 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) |
7 | rngcval.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
8 | 7 | sqxpeqd 5587 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) |
9 | 8 | eqcomd 2827 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵)) |
10 | 6, 9 | sylan9eqr 2878 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵)) |
11 | 10 | reseq2d 5853 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHomo ↾ (𝐵 × 𝐵))) |
12 | rngcval.h | . . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
13 | 12 | eqcomd 2827 | . . . . . 6 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻) |
14 | 13 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻) |
15 | 11, 14 | eqtrd 2856 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻) |
16 | 4, 15 | oveq12d 7174 | . . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
17 | rngcval.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
18 | 17 | elexd 3514 | . . 3 ⊢ (𝜑 → 𝑈 ∈ V) |
19 | ovexd 7191 | . . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V) | |
20 | 2, 16, 18, 19 | fvmptd2 6776 | . 2 ⊢ (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
21 | 1, 20 | syl5eq 2868 | 1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 × cxp 5553 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 ↾cat cresc 17078 ExtStrCatcestrc 17372 Rngcrng 44165 RngHomo crngh 44176 RngCatcrngc 44248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-rngc 44250 |
This theorem is referenced by: rngcbas 44256 rngchomfval 44257 rngccofval 44261 dfrngc2 44263 rngccat 44269 rngcid 44270 rngcifuestrc 44288 funcrngcsetc 44289 |
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