![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
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 47366 | . . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) |
6 | simprl 770 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑣 = 〈𝑋, 𝑌〉) | |
7 | 6 | fveq2d 6901 | . . . . . 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 8006 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
11 | 8, 9, 10 | syl2anc 583 | . . . . . . 7 ⊢ (𝜑 → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
13 | 7, 12 | eqtrd 2768 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌) |
14 | simprr 772 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
15 | 13, 14 | oveq12d 7438 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) RingHom 𝑧) = (𝑌 RingHom 𝑍)) |
16 | 6 | fveq2d 6901 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘〈𝑋, 𝑌〉)) |
17 | op1stg 8005 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) | |
18 | 8, 9, 17 | syl2anc 583 | . . . . . . 7 ⊢ (𝜑 → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
19 | 18 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
20 | 16, 19 | eqtrd 2768 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋) |
21 | 20, 13 | oveq12d 7438 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((1st ‘𝑣) RingHom (2nd ‘𝑣)) = (𝑋 RingHom 𝑌)) |
22 | eqidd 2729 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓)) | |
23 | 15, 21, 22 | mpoeq123dv 7495 | . . 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 5715 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
25 | 8, 9, 24 | syl2anc 583 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
26 | ringccoALTV.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
27 | ovex 7453 | . . . . 5 ⊢ (𝑌 RingHom 𝑍) ∈ V | |
28 | ovex 7453 | . . . . 5 ⊢ (𝑋 RingHom 𝑌) ∈ V | |
29 | 27, 28 | mpoex 8084 | . . . 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 7573 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔 ∘ 𝑓))) |
32 | simprl 770 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺) | |
33 | simprr 772 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹) | |
34 | 32, 33 | coeq12d 5867 | . 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 7937 | . . 3 ⊢ ((𝐺 ∈ (𝑌 RingHom 𝑍) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌)) → (𝐺 ∘ 𝐹) ∈ V) | |
38 | 35, 36, 37 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
39 | 31, 34, 35, 36, 38 | ovmpod 7573 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 〈cop 4635 × cxp 5676 ∘ ccom 5682 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 1st c1st 7991 2nd c2nd 7992 Basecbs 17180 compcco 17245 RingHom crh 20408 RingCatALTVcringcALTV 47349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-hom 17257 df-cco 17258 df-ringcALTV 47350 |
This theorem is referenced by: ringccatidALTV 47368 ringcsectALTV 47371 funcringcsetclem9ALTV 47383 |
Copyright terms: Public domain | W3C validator |