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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngchomffvalALTV | Structured version Visualization version GIF version |
Description: The value of the functionalized Hom-set operation in the category of non-unital rings (in a universe) in maps-to notation for an operation. (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngchomffvalALTV.c | ⊢ 𝐶 = (RngCatALTV‘𝑈) |
rngchomffvalALTV.b | ⊢ 𝐵 = (Base‘𝐶) |
rngchomffvalALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngchomffvalALTV.h | ⊢ 𝐹 = (Homf ‘𝐶) |
Ref | Expression |
---|---|
rngchomffvalALTV | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngchomffvalALTV.c | . . . 4 ⊢ 𝐶 = (RngCatALTV‘𝑈) | |
2 | rngchomffvalALTV.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | rngchomffvalALTV.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | eqid 2825 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | 1, 2, 3, 4 | rngchomfvalALTV 42831 | . . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦))) |
6 | eqid 2825 | . . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦)) | |
7 | ovex 6937 | . . . . 5 ⊢ (𝑥 RngHomo 𝑦) ∈ V | |
8 | 6, 7 | fnmpt2i 7502 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦)) Fn (𝐵 × 𝐵) |
9 | fneq1 6212 | . . . 4 ⊢ ((Hom ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦)) → ((Hom ‘𝐶) Fn (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦)) Fn (𝐵 × 𝐵))) | |
10 | 8, 9 | mpbiri 250 | . . 3 ⊢ ((Hom ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦)) → (Hom ‘𝐶) Fn (𝐵 × 𝐵)) |
11 | rngchomffvalALTV.h | . . . 4 ⊢ 𝐹 = (Homf ‘𝐶) | |
12 | 11, 2, 4 | fnhomeqhomf 16703 | . . 3 ⊢ ((Hom ‘𝐶) Fn (𝐵 × 𝐵) → 𝐹 = (Hom ‘𝐶)) |
13 | 5, 10, 12 | 3syl 18 | . 2 ⊢ (𝜑 → 𝐹 = (Hom ‘𝐶)) |
14 | 13, 5 | eqtrd 2861 | 1 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 × cxp 5340 Fn wfn 6118 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 Basecbs 16222 Hom chom 16316 Homf chomf 16679 RngHomo crngh 42732 RngCatALTVcrngcALTV 42805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-hom 16329 df-cco 16330 df-homf 16683 df-rngcALTV 42807 |
This theorem is referenced by: rngchomrnghmresALTV 42843 |
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