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 41779 | . . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) |
6 | simprl 790 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑣 = 〈𝑋, 𝑌〉) | |
7 | 6 | fveq2d 6107 | . . . . . 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 7072 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
11 | 8, 9, 10 | syl2anc 691 | . . . . . . 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 2644 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌) |
14 | simprr 792 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
15 | 13, 14 | oveq12d 6567 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) RngHomo 𝑧) = (𝑌 RngHomo 𝑍)) |
16 | 6 | fveq2d 6107 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘〈𝑋, 𝑌〉)) |
17 | op1stg 7071 | . . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘〈𝑋, 𝑌〉) = 𝑋) | |
18 | 8, 9, 17 | syl2anc 691 | . . . . . . 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 2644 | . . . . 5 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋) |
21 | 20, 13 | oveq12d 6567 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) = (𝑋 RngHomo 𝑌)) |
22 | eqidd 2611 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓)) | |
23 | 15, 21, 22 | mpt2eq123dv 6615 | . . 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 5072 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
25 | 8, 9, 24 | syl2anc 691 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
26 | rngccoALTV.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
27 | ovex 6577 | . . . . 5 ⊢ (𝑌 RngHomo 𝑍) ∈ V | |
28 | ovex 6577 | . . . . 5 ⊢ (𝑋 RngHomo 𝑌) ∈ V | |
29 | 27, 28 | mpt2ex 7136 | . . . 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 6686 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (𝑔 ∈ (𝑌 RngHomo 𝑍), 𝑓 ∈ (𝑋 RngHomo 𝑌) ↦ (𝑔 ∘ 𝑓))) |
32 | simprl 790 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺) | |
33 | simprr 792 | . . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹) | |
34 | 32, 33 | coeq12d 5208 | . 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 7010 | . . 3 ⊢ ((𝐺 ∈ (𝑌 RngHomo 𝑍) ∧ 𝐹 ∈ (𝑋 RngHomo 𝑌)) → (𝐺 ∘ 𝐹) ∈ V) | |
38 | 35, 36, 37 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
39 | 31, 34, 35, 36, 38 | ovmpt2d 6686 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 × cxp 5036 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 Basecbs 15695 compcco 15780 RngHomo crngh 41675 RngCatALTVcrngcALTV 41750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-hom 15793 df-cco 15794 df-rngcALTV 41752 |
This theorem is referenced by: rngccatidALTV 41781 rngcsectALTV 41784 rhmsubcALTVlem4 41900 |
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