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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 47307. (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 47302 | . . . 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 2727 | . . . 4 β’ (Idβπ ) = (Idβπ ) | |
11 | 5 | adantr 480 | . . . 4 β’ ((π β§ π β π΅) β π β WUni) |
12 | simpr 484 | . . . 4 β’ ((π β§ π β π΅) β π β π΅) | |
13 | eqid 2727 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
14 | 1, 3, 10, 11, 12, 13 | ringcidALTV 47293 | . . 3 β’ ((π β§ π β π΅) β ((Idβπ )βπ) = ( I βΎ (Baseβπ))) |
15 | 9, 14 | fveq12d 6898 | . 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 47285 | . . . . . 6 β’ (π β π΅ = (π β© Ring)) |
17 | 16 | eleq2d 2814 | . . . . 5 β’ (π β (π β π΅ β π β (π β© Ring))) |
18 | elin 3960 | . . . . . 6 β’ (π β (π β© Ring) β (π β π β§ π β Ring)) | |
19 | 18 | simprbi 496 | . . . . 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 406 | . . 3 β’ ((π β§ π β π΅) β π β Ring) |
22 | 13 | idrhm 20418 | . . 3 β’ (π β Ring β ( I βΎ (Baseβπ)) β (π RingHom π)) |
23 | fvresi 7176 | . . 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 47298 | . . . 4 β’ ((π β§ π β π΅) β (πΉβπ) = (Baseβπ)) |
26 | 25 | fveq2d 6895 | . . 3 β’ ((π β§ π β π΅) β ((Idβπ)β(πΉβπ)) = ((Idβπ)β(Baseβπ))) |
27 | eqid 2727 | . . . 4 β’ (Idβπ) = (Idβπ) | |
28 | 1, 3, 5 | ringcbasbasALTV 47297 | . . . 4 β’ ((π β§ π β π΅) β (Baseβπ) β π) |
29 | 2, 27, 11, 28 | setcid 18066 | . . 3 β’ ((π β§ π β π΅) β ((Idβπ)β(Baseβπ)) = ( I βΎ (Baseβπ))) |
30 | 26, 29 | eqtr2d 2768 | . 2 β’ ((π β§ π β π΅) β ( I βΎ (Baseβπ)) = ((Idβπ)β(πΉβπ))) |
31 | 15, 24, 30 | 3eqtrd 2771 | 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 395 = wceq 1534 β wcel 2099 β© cin 3943 β¦ cmpt 5225 I cid 5569 βΎ cres 5674 βcfv 6542 (class class class)co 7414 β cmpo 7416 WUnicwun 10715 Basecbs 17171 Idccid 17636 SetCatcsetc 18055 Ringcrg 20164 RingHom crh 20397 RingCatALTVcringcALTV 47272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-wun 10717 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-plusg 17237 df-hom 17248 df-cco 17249 df-0g 17414 df-cat 17639 df-cid 17640 df-setc 18056 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-grp 18884 df-ghm 19159 df-mgp 20066 df-ur 20113 df-ring 20166 df-rhm 20400 df-ringcALTV 47273 |
This theorem is referenced by: funcringcsetcALTV 47307 |
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