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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngchom | Structured version Visualization version GIF version |
Description: Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) |
Ref | Expression |
---|---|
rngcbas.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rngcbas.b | ⊢ 𝐵 = (Base‘𝐶) |
rngcbas.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngchomfval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
rngchom.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
rngchom.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
rngchom | ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RngHomo 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcbas.c | . . . 4 ⊢ 𝐶 = (RngCat‘𝑈) | |
2 | rngcbas.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | rngcbas.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
4 | rngchomfval.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
5 | 1, 2, 3, 4 | rngchomfval 43601 | . . 3 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
6 | 5 | oveqd 6987 | . 2 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌)) |
7 | rngchom.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | rngchom.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | 7, 8 | ovresd 7125 | . 2 ⊢ (𝜑 → (𝑋( RngHomo ↾ (𝐵 × 𝐵))𝑌) = (𝑋 RngHomo 𝑌)) |
10 | 6, 9 | eqtrd 2808 | 1 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RngHomo 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 × cxp 5399 ↾ cres 5403 ‘cfv 6182 (class class class)co 6970 Basecbs 16333 Hom chom 16426 RngHomo crngh 43520 RngCatcrngc 43592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-z 11788 df-dec 11906 df-uz 12053 df-fz 12703 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-hom 16439 df-cco 16440 df-resc 16933 df-estrc 17225 df-rnghomo 43522 df-rngc 43594 |
This theorem is referenced by: elrngchom 43603 rnghmsubcsetclem1 43610 rngcsect 43615 funcrngcsetc 43633 zrinitorngc 43635 zrtermorngc 43636 |
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