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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 20662 | . . 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 4220 | . . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring)) | |
6 | 5 | sqxpeqd 5720 | . . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) |
7 | ringcval.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) | |
8 | 7 | sqxpeqd 5720 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring))) |
9 | 8 | eqcomd 2740 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = (𝐵 × 𝐵)) |
10 | 6, 9 | sylan9eqr 2796 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = (𝐵 × 𝐵)) |
11 | 10 | reseq2d 5999 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = ( RingHom ↾ (𝐵 × 𝐵))) |
12 | ringcval.h | . . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | |
13 | 12 | eqcomd 2740 | . . . . . 6 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻) |
14 | 13 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻) |
15 | 11, 14 | eqtrd 2774 | . . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = 𝐻) |
16 | 4, 15 | oveq12d 7448 | . . 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 3501 | . . 3 ⊢ (𝜑 → 𝑈 ∈ V) |
19 | ovexd 7465 | . . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V) | |
20 | 2, 16, 18, 19 | fvmptd2 7023 | . 2 ⊢ (𝜑 → (RingCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
21 | 1, 20 | eqtrid 2786 | 1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 × cxp 5686 ↾ cres 5690 ‘cfv 6562 (class class class)co 7430 ↾cat cresc 17855 ExtStrCatcestrc 18176 Ringcrg 20250 RingHom crh 20485 RingCatcringc 20661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-res 5700 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-ringc 20662 |
This theorem is referenced by: ringcbas 20666 ringchomfval 20667 ringccofval 20671 dfringc2 20673 ringccat 20679 ringcid 20680 funcringcsetc 20690 |
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