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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > funcringcsetclem7ALTV | Structured version Visualization version GIF version |
Description: Lemma 7 for funcringcsetcALTV 46440. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
funcringcsetcALTV.r | β’ π = (RingCatALTVβπ) |
funcringcsetcALTV.s | β’ π = (SetCatβπ) |
funcringcsetcALTV.b | β’ π΅ = (Baseβπ ) |
funcringcsetcALTV.c | β’ πΆ = (Baseβπ) |
funcringcsetcALTV.u | β’ (π β π β WUni) |
funcringcsetcALTV.f | β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) |
funcringcsetcALTV.g | β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦)))) |
Ref | Expression |
---|---|
funcringcsetclem7ALTV | β’ ((π β§ π β π΅) β ((ππΊπ)β((Idβπ )βπ)) = ((Idβπ)β(πΉβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcringcsetcALTV.r | . . . . 5 β’ π = (RingCatALTVβπ) | |
2 | funcringcsetcALTV.s | . . . . 5 β’ π = (SetCatβπ) | |
3 | funcringcsetcALTV.b | . . . . 5 β’ π΅ = (Baseβπ ) | |
4 | funcringcsetcALTV.c | . . . . 5 β’ πΆ = (Baseβπ) | |
5 | funcringcsetcALTV.u | . . . . 5 β’ (π β π β WUni) | |
6 | funcringcsetcALTV.f | . . . . 5 β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) | |
7 | funcringcsetcALTV.g | . . . . 5 β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦)))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | funcringcsetclem5ALTV 46435 | . . . 4 β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΊπ) = ( I βΎ (π RingHom π))) |
9 | 8 | anabsan2 673 | . . 3 β’ ((π β§ π β π΅) β (ππΊπ) = ( I βΎ (π RingHom π))) |
10 | eqid 2737 | . . . 4 β’ (Idβπ ) = (Idβπ ) | |
11 | 5 | adantr 482 | . . . 4 β’ ((π β§ π β π΅) β π β WUni) |
12 | simpr 486 | . . . 4 β’ ((π β§ π β π΅) β π β π΅) | |
13 | eqid 2737 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
14 | 1, 3, 10, 11, 12, 13 | ringcidALTV 46426 | . . 3 β’ ((π β§ π β π΅) β ((Idβπ )βπ) = ( I βΎ (Baseβπ))) |
15 | 9, 14 | fveq12d 6854 | . 2 β’ ((π β§ π β π΅) β ((ππΊπ)β((Idβπ )βπ)) = (( I βΎ (π RingHom π))β( I βΎ (Baseβπ)))) |
16 | 1, 3, 5 | ringcbasALTV 46418 | . . . . . 6 β’ (π β π΅ = (π β© Ring)) |
17 | 16 | eleq2d 2824 | . . . . 5 β’ (π β (π β π΅ β π β (π β© Ring))) |
18 | elin 3931 | . . . . . 6 β’ (π β (π β© Ring) β (π β π β§ π β Ring)) | |
19 | 18 | simprbi 498 | . . . . 5 β’ (π β (π β© Ring) β π β Ring) |
20 | 17, 19 | syl6bi 253 | . . . 4 β’ (π β (π β π΅ β π β Ring)) |
21 | 20 | imp 408 | . . 3 β’ ((π β§ π β π΅) β π β Ring) |
22 | 13 | idrhm 20172 | . . 3 β’ (π β Ring β ( I βΎ (Baseβπ)) β (π RingHom π)) |
23 | fvresi 7124 | . . 3 β’ (( I βΎ (Baseβπ)) β (π RingHom π) β (( I βΎ (π RingHom π))β( I βΎ (Baseβπ))) = ( I βΎ (Baseβπ))) | |
24 | 21, 22, 23 | 3syl 18 | . 2 β’ ((π β§ π β π΅) β (( I βΎ (π RingHom π))β( I βΎ (Baseβπ))) = ( I βΎ (Baseβπ))) |
25 | 1, 2, 3, 4, 5, 6 | funcringcsetclem1ALTV 46431 | . . . 4 β’ ((π β§ π β π΅) β (πΉβπ) = (Baseβπ)) |
26 | 25 | fveq2d 6851 | . . 3 β’ ((π β§ π β π΅) β ((Idβπ)β(πΉβπ)) = ((Idβπ)β(Baseβπ))) |
27 | eqid 2737 | . . . 4 β’ (Idβπ) = (Idβπ) | |
28 | 1, 3, 5 | ringcbasbasALTV 46430 | . . . 4 β’ ((π β§ π β π΅) β (Baseβπ) β π) |
29 | 2, 27, 11, 28 | setcid 17979 | . . 3 β’ ((π β§ π β π΅) β ((Idβπ)β(Baseβπ)) = ( I βΎ (Baseβπ))) |
30 | 26, 29 | eqtr2d 2778 | . 2 β’ ((π β§ π β π΅) β ( I βΎ (Baseβπ)) = ((Idβπ)β(πΉβπ))) |
31 | 15, 24, 30 | 3eqtrd 2781 | 1 β’ ((π β§ π β π΅) β ((ππΊπ)β((Idβπ )βπ)) = ((Idβπ)β(πΉβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β© cin 3914 β¦ cmpt 5193 I cid 5535 βΎ cres 5640 βcfv 6501 (class class class)co 7362 β cmpo 7364 WUnicwun 10643 Basecbs 17090 Idccid 17552 SetCatcsetc 17968 Ringcrg 19971 RingHom crh 20152 RingCatALTVcringcALTV 46376 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-wun 10645 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-plusg 17153 df-hom 17164 df-cco 17165 df-0g 17330 df-cat 17555 df-cid 17556 df-setc 17969 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-grp 18758 df-ghm 19013 df-mgp 19904 df-ur 19921 df-ring 19973 df-rnghom 20155 df-ringcALTV 46378 |
This theorem is referenced by: funcringcsetcALTV 46440 |
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