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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringcco | Structured version Visualization version GIF version |
Description: Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
Ref | Expression |
---|---|
ringcco.c | ⊢ 𝐶 = (RingCat‘𝑈) |
ringcco.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
ringcco.o | ⊢ · = (comp‘𝐶) |
ringcco.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
ringcco.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
ringcco.z | ⊢ (𝜑 → 𝑍 ∈ 𝑈) |
ringcco.f | ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)) |
ringcco.g | ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍)) |
Ref | Expression |
---|---|
ringcco | ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcco.c | . . . . 5 ⊢ 𝐶 = (RingCat‘𝑈) | |
2 | ringcco.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | ringcco.o | . . . . 5 ⊢ · = (comp‘𝐶) | |
4 | 1, 2, 3 | ringccofval 46816 | . . . 4 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) |
5 | 4 | oveqd 7421 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉 · 𝑍) = (〈𝑋, 𝑌〉(comp‘(ExtStrCat‘𝑈))𝑍)) |
6 | 5 | oveqd 7421 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉(comp‘(ExtStrCat‘𝑈))𝑍)𝐹)) |
7 | eqid 2733 | . . 3 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
8 | eqid 2733 | . . 3 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)) | |
9 | ringcco.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
10 | ringcco.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
11 | ringcco.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
12 | eqid 2733 | . . 3 ⊢ (Base‘𝑋) = (Base‘𝑋) | |
13 | eqid 2733 | . . 3 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
14 | eqid 2733 | . . 3 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
15 | ringcco.f | . . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)) | |
16 | ringcco.g | . . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍)) | |
17 | 7, 2, 8, 9, 10, 11, 12, 13, 14, 15, 16 | estrcco 18077 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉(comp‘(ExtStrCat‘𝑈))𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
18 | 6, 17 | eqtrd 2773 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 〈cop 4633 ∘ ccom 5679 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 compcco 17205 ExtStrCatcestrc 18069 RingCatcringc 46803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-hom 17217 df-cco 17218 df-0g 17383 df-resc 17754 df-estrc 18070 df-mhm 18667 df-ghm 19084 df-mgp 19980 df-ur 19997 df-ring 20049 df-rnghom 20240 df-ringc 46805 |
This theorem is referenced by: ringcsect 46831 funcringcsetcALTV2lem9 46844 |
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