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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngccoALTV | Structured version Visualization version GIF version |
Description: Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
Ref | Expression |
---|---|
rngcbasALTV.c | ⊢ 𝐶 = (RngCatALTV‘𝑈) |
rngcbasALTV.b | ⊢ 𝐵 = (Base‘𝐶) |
rngcbasALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngccofvalALTV.o | ⊢ · = (comp‘𝐶) |
rngccoALTV.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
rngccoALTV.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
rngccoALTV.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
rngccoALTV.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋 RngHomo 𝑌)) |
rngccoALTV.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌 RngHomo 𝑍)) |
Ref | Expression |
---|---|
rngccoALTV | ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcbasALTV.c | . . . 4 ⊢ 𝐶 = (RngCatALTV‘𝑈) | |
2 | rngcbasALTV.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | rngcbasALTV.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | rngccofvalALTV.o | . . . 4 ⊢ · = (comp‘𝐶) | |
5 | 1, 2, 3, 4 | rngccofvalALTV 42515 | . . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) |
6 | simprl 754 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑣 = 〈𝑋, 𝑌〉) | |
7 | 6 | fveq2d 6336 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = (2nd ‘〈𝑋, 𝑌〉)) |
8 | rngccoALTV.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | rngccoALTV.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | op2ndg 7328 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
11 | 8, 9, 10 | syl2anc 573 | . . . . . . 7 ⊢ (𝜑 → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
12 | 11 | adantr 466 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
13 | 7, 12 | eqtrd 2805 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌) |
14 | simprr 756 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
15 | 13, 14 | oveq12d 6811 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) RngHomo 𝑧) = (𝑌 RngHomo 𝑍)) |
16 | 6 | fveq2d 6336 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘〈𝑋, 𝑌〉)) |
17 | op1stg 7327 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) | |
18 | 8, 9, 17 | syl2anc 573 | . . . . . . 7 ⊢ (𝜑 → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
19 | 18 | adantr 466 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) |
20 | 16, 19 | eqtrd 2805 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋) |
21 | 20, 13 | oveq12d 6811 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) = (𝑋 RngHomo 𝑌)) |
22 | eqidd 2772 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓)) | |
23 | 15, 21, 22 | mpt2eq123dv 6864 | . . 3 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑌 RngHomo 𝑍), 𝑓 ∈ (𝑋 RngHomo 𝑌) ↦ (𝑔 ∘ 𝑓))) |
24 | opelxpi 5288 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
25 | 8, 9, 24 | syl2anc 573 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
26 | rngccoALTV.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
27 | ovex 6823 | . . . . 5 ⊢ (𝑌 RngHomo 𝑍) ∈ V | |
28 | ovex 6823 | . . . . 5 ⊢ (𝑋 RngHomo 𝑌) ∈ V | |
29 | 27, 28 | mpt2ex 7397 | . . . 4 ⊢ (𝑔 ∈ (𝑌 RngHomo 𝑍), 𝑓 ∈ (𝑋 RngHomo 𝑌) ↦ (𝑔 ∘ 𝑓)) ∈ V |
30 | 29 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑌 RngHomo 𝑍), 𝑓 ∈ (𝑋 RngHomo 𝑌) ↦ (𝑔 ∘ 𝑓)) ∈ V) |
31 | 5, 23, 25, 26, 30 | ovmpt2d 6935 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (𝑔 ∈ (𝑌 RngHomo 𝑍), 𝑓 ∈ (𝑋 RngHomo 𝑌) ↦ (𝑔 ∘ 𝑓))) |
32 | simprl 754 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺) | |
33 | simprr 756 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹) | |
34 | 32, 33 | coeq12d 5425 | . 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹)) |
35 | rngccoALTV.g | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌 RngHomo 𝑍)) | |
36 | rngccoALTV.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋 RngHomo 𝑌)) | |
37 | coexg 7264 | . . 3 ⊢ ((𝐺 ∈ (𝑌 RngHomo 𝑍) ∧ 𝐹 ∈ (𝑋 RngHomo 𝑌)) → (𝐺 ∘ 𝐹) ∈ V) | |
38 | 35, 36, 37 | syl2anc 573 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
39 | 31, 34, 35, 36, 38 | ovmpt2d 6935 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 〈cop 4322 × cxp 5247 ∘ ccom 5253 ‘cfv 6031 (class class class)co 6793 ↦ cmpt2 6795 1st c1st 7313 2nd c2nd 7314 Basecbs 16064 compcco 16161 RngHomo crngh 42413 RngCatALTVcrngcALTV 42486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-hom 16174 df-cco 16175 df-rngcALTV 42488 |
This theorem is referenced by: rngccatidALTV 42517 rngcsectALTV 42520 rhmsubcALTVlem4 42635 |
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