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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 43730 | . . 3 ⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))) | |
3 | fveq2 6545 | . . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈)) | |
4 | 3 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈)) |
5 | ineq1 4107 | . . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng)) | |
6 | 5 | sqxpeqd 5482 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) |
7 | rngcval.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | |
8 | 7 | sqxpeqd 5482 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng))) |
9 | 8 | eqcomd 2803 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵)) |
10 | 6, 9 | sylan9eqr 2855 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵)) |
11 | 10 | reseq2d 5741 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHomo ↾ (𝐵 × 𝐵))) |
12 | rngcval.h | . . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
13 | 12 | eqcomd 2803 | . . . . . 6 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻) |
14 | 13 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻) |
15 | 11, 14 | eqtrd 2833 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻) |
16 | 4, 15 | oveq12d 7041 | . . 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 3460 | . . 3 ⊢ (𝜑 → 𝑈 ∈ V) |
19 | ovexd 7057 | . . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V) | |
20 | 2, 16, 18, 19 | fvmptd2 6649 | . 2 ⊢ (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
21 | 1, 20 | syl5eq 2845 | 1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 Vcvv 3440 ∩ cin 3864 × cxp 5448 ↾ cres 5452 ‘cfv 6232 (class class class)co 7023 ↾cat cresc 16911 ExtStrCatcestrc 17205 Rngcrng 43645 RngHomo crngh 43656 RngCatcrngc 43728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-res 5462 df-iota 6196 df-fun 6234 df-fv 6240 df-ov 7026 df-rngc 43730 |
This theorem is referenced by: rngcbas 43736 rngchomfval 43737 rngccofval 43741 dfrngc2 43743 rngccat 43749 rngcid 43750 rngcifuestrc 43768 funcrngcsetc 43769 |
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