Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringchomfvalALTV | Structured version Visualization version GIF version |
Description: Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringcbasALTV.c | ⊢ 𝐶 = (RingCatALTV‘𝑈) |
ringcbasALTV.b | ⊢ 𝐵 = (Base‘𝐶) |
ringcbasALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
ringchomfvalALTV.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
ringchomfvalALTV | ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringchomfvalALTV.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
2 | ringcbasALTV.c | . . . . 5 ⊢ 𝐶 = (RingCatALTV‘𝑈) | |
3 | ringcbasALTV.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | ringcbasALTV.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 2, 4, 3 | ringcbasALTV 45138 | . . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) |
6 | eqidd 2739 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))) | |
7 | eqidd 2739 | . . . . 5 ⊢ (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))) | |
8 | 2, 3, 5, 6, 7 | ringcvalALTV 45099 | . . . 4 ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))〉}) |
9 | 8 | fveq2d 6678 | . . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))〉})) |
10 | 1, 9 | syl5eq 2785 | . 2 ⊢ (𝜑 → 𝐻 = (Hom ‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))〉})) |
11 | 4 | fvexi 6688 | . . . 4 ⊢ 𝐵 ∈ V |
12 | 11, 11 | mpoex 7803 | . . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)) ∈ V |
13 | catstr 17332 | . . . 4 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))〉} Struct 〈1, ;15〉 | |
14 | homid 16791 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
15 | snsstp2 4705 | . . . 4 ⊢ {〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))〉} ⊆ {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))〉} | |
16 | 13, 14, 15 | strfv 16634 | . . 3 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)) ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)) = (Hom ‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))〉})) |
17 | 12, 16 | mp1i 13 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)) = (Hom ‘{〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑔 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑓 ∘ 𝑔)))〉})) |
18 | 10, 17 | eqtr4d 2776 | 1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3398 {ctp 4520 〈cop 4522 × cxp 5523 ∘ ccom 5529 ‘cfv 6339 (class class class)co 7170 ∈ cmpo 7172 1st c1st 7712 2nd c2nd 7713 1c1 10616 5c5 11774 ;cdc 12179 ndxcnx 16583 Basecbs 16586 Hom chom 16679 compcco 16680 RingHom crh 19586 RingCatALTVcringcALTV 45096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-hom 16692 df-cco 16693 df-ringcALTV 45098 |
This theorem is referenced by: ringchomALTV 45140 ringccofvalALTV 45142 |
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