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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngccofvalALTV | 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) |
Ref | Expression |
---|---|
rngccofvalALTV | ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcbasALTV.c | . . . 4 ⊢ 𝐶 = (RngCatALTV‘𝑈) | |
2 | rngcbasALTV.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
3 | rngcbasALTV.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
4 | 1, 3, 2 | rngcbasALTV 44248 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
5 | eqid 2821 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
6 | 1, 3, 2, 5 | rngchomfvalALTV 44249 | . . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦))) |
7 | eqidd 2822 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) | |
8 | 1, 2, 4, 6, 7 | rngcvalALTV 44226 | . . 3 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉}) |
9 | 8 | fveq2d 6668 | . 2 ⊢ (𝜑 → (comp‘𝐶) = (comp‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉})) |
10 | rngccofvalALTV.o | . 2 ⊢ · = (comp‘𝐶) | |
11 | 3 | fvexi 6678 | . . . . 5 ⊢ 𝐵 ∈ V |
12 | sqxpexg 7471 | . . . . 5 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ (𝐵 × 𝐵) ∈ V |
14 | 13, 11 | mpoex 7771 | . . 3 ⊢ (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) ∈ V |
15 | catstr 17221 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉} Struct 〈1, ;15〉 | |
16 | ccoid 16684 | . . . 4 ⊢ comp = Slot (comp‘ndx) | |
17 | snsstp3 4744 | . . . 4 ⊢ {〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉} ⊆ {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉} | |
18 | 15, 16, 17 | strfv 16525 | . . 3 ⊢ ((𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) ∈ V → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (comp‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉})) |
19 | 14, 18 | ax-mp 5 | . 2 ⊢ (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (comp‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (Hom ‘𝐶)〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉}) |
20 | 9, 10, 19 | 3eqtr4g 2881 | 1 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {ctp 4564 〈cop 4566 × cxp 5547 ∘ ccom 5553 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 1st c1st 7681 2nd c2nd 7682 1c1 10532 5c5 11689 ;cdc 12092 ndxcnx 16474 Basecbs 16477 Hom chom 16570 compcco 16571 RngHomo crngh 44150 RngCatALTVcrngcALTV 44223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-hom 16583 df-cco 16584 df-rngcALTV 44225 |
This theorem is referenced by: rngccoALTV 44253 |
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