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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringccofval | 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) |
Ref | Expression |
---|---|
ringccofval | ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcco.c | . . . 4 ⊢ 𝐶 = (RingCat‘𝑈) | |
2 | ringcco.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | eqid 2738 | . . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
4 | 1, 3, 2 | ringcbas 46100 | . . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Ring)) |
5 | eqid 2738 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | 1, 3, 2, 5 | ringchomfval 46101 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = ( RingHom ↾ ((Base‘𝐶) × (Base‘𝐶)))) |
7 | 1, 2, 4, 6 | ringcval 46097 | . . 3 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶))) |
8 | 7 | fveq2d 6842 | . 2 ⊢ (𝜑 → (comp‘𝐶) = (comp‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)))) |
9 | ringcco.o | . . 3 ⊢ · = (comp‘𝐶) | |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → · = (comp‘𝐶)) |
11 | eqid 2738 | . . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)) | |
12 | eqid 2738 | . . 3 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈)) | |
13 | fvexd 6853 | . . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V) | |
14 | 4, 6 | rhmresfn 46098 | . . 3 ⊢ (𝜑 → (Hom ‘𝐶) Fn ((Base‘𝐶) × (Base‘𝐶))) |
15 | inss1 4187 | . . . . 5 ⊢ (𝑈 ∩ Ring) ⊆ 𝑈 | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑈 ∩ Ring) ⊆ 𝑈) |
17 | eqid 2738 | . . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈) | |
18 | 17, 2 | estrcbas 17948 | . . . . 5 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈))) |
19 | 18 | eqcomd 2744 | . . . 4 ⊢ (𝜑 → (Base‘(ExtStrCat‘𝑈)) = 𝑈) |
20 | 16, 4, 19 | 3sstr4d 3990 | . . 3 ⊢ (𝜑 → (Base‘𝐶) ⊆ (Base‘(ExtStrCat‘𝑈))) |
21 | eqid 2738 | . . 3 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈)) | |
22 | 11, 12, 13, 14, 20, 21 | rescco 17652 | . 2 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (comp‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝐶)))) |
23 | 8, 10, 22 | 3eqtr4d 2788 | 1 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∩ cin 3908 ⊆ wss 3909 ‘cfv 6492 (class class class)co 7350 Basecbs 17019 Hom chom 17080 compcco 17081 ↾cat cresc 17627 ExtStrCatcestrc 17945 Ringcrg 19894 RingCatcringc 46092 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12553 df-uz 12698 df-fz 13355 df-struct 16955 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-hom 17093 df-cco 17094 df-0g 17259 df-resc 17630 df-estrc 17946 df-mhm 18537 df-ghm 18941 df-mgp 19832 df-ur 19849 df-ring 19896 df-rnghom 20075 df-ringc 46094 |
This theorem is referenced by: ringcco 46106 |
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