Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringccoALTV | Structured version Visualization version GIF version |
Description: Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringcbasALTV.c | ⊢ 𝐶 = (RingCatALTV‘𝑈) |
ringcbasALTV.b | ⊢ 𝐵 = (Base‘𝐶) |
ringcbasALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
ringccoALTV.o | ⊢ · = (comp‘𝐶) |
ringccoALTV.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringccoALTV.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ringccoALTV.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
ringccoALTV.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋 RingHom 𝑌)) |
ringccoALTV.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌 RingHom 𝑍)) |
Ref | Expression |
---|---|
ringccoALTV | ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcbasALTV.c | . . . 4 ⊢ 𝐶 = (RingCatALTV‘𝑈) | |
2 | ringcbasALTV.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | ringcbasALTV.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | ringccoALTV.o | . . . 4 ⊢ · = (comp‘𝐶) | |
5 | 1, 2, 3, 4 | ringccofvalALTV 45235 | . . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) |
6 | simprl 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑣 = 〈𝑋, 𝑌〉) | |
7 | 6 | fveq2d 6710 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = (2nd ‘〈𝑋, 𝑌〉)) |
8 | ringccoALTV.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | ringccoALTV.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | op2ndg 7763 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
11 | 8, 9, 10 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
12 | 11 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
13 | 7, 12 | eqtrd 2774 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌) |
14 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
15 | 13, 14 | oveq12d 7220 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) RingHom 𝑧) = (𝑌 RingHom 𝑍)) |
16 | 6 | fveq2d 6710 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘〈𝑋, 𝑌〉)) |
17 | op1stg 7762 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) | |
18 | 8, 9, 17 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
19 | 18 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
20 | 16, 19 | eqtrd 2774 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋) |
21 | 20, 13 | oveq12d 7220 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((1st ‘𝑣) RingHom (2nd ‘𝑣)) = (𝑋 RingHom 𝑌)) |
22 | eqidd 2735 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓)) | |
23 | 15, 21, 22 | mpoeq123dv 7275 | . . 3 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔 ∘ 𝑓))) |
24 | opelxpi 5577 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
25 | 8, 9, 24 | syl2anc 587 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
26 | ringccoALTV.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
27 | ovex 7235 | . . . . 5 ⊢ (𝑌 RingHom 𝑍) ∈ V | |
28 | ovex 7235 | . . . . 5 ⊢ (𝑋 RingHom 𝑌) ∈ V | |
29 | 27, 28 | mpoex 7839 | . . . 4 ⊢ (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔 ∘ 𝑓)) ∈ V |
30 | 29 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔 ∘ 𝑓)) ∈ V) |
31 | 5, 23, 25, 26, 30 | ovmpod 7350 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔 ∘ 𝑓))) |
32 | simprl 771 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺) | |
33 | simprr 773 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹) | |
34 | 32, 33 | coeq12d 5722 | . 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹)) |
35 | ringccoALTV.g | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌 RingHom 𝑍)) | |
36 | ringccoALTV.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋 RingHom 𝑌)) | |
37 | coexg 7696 | . . 3 ⊢ ((𝐺 ∈ (𝑌 RingHom 𝑍) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌)) → (𝐺 ∘ 𝐹) ∈ V) | |
38 | 35, 36, 37 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
39 | 31, 34, 35, 36, 38 | ovmpod 7350 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3401 〈cop 4537 × cxp 5538 ∘ ccom 5544 ‘cfv 6369 (class class class)co 7202 ∈ cmpo 7204 1st c1st 7748 2nd c2nd 7749 Basecbs 16684 compcco 16779 RingHom crh 19704 RingCatALTVcringcALTV 45189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-hom 16791 df-cco 16792 df-ringcALTV 45191 |
This theorem is referenced by: ringccatidALTV 45237 ringcsectALTV 45240 funcringcsetclem9ALTV 45252 |
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