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Mirrors > Home > MPE Home > Th. List > Mathboxes > funcringcsetcALTV2lem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for funcringcsetcALTV2 46417. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
funcringcsetcALTV2.r | β’ π = (RingCatβπ) |
funcringcsetcALTV2.s | β’ π = (SetCatβπ) |
funcringcsetcALTV2.b | β’ π΅ = (Baseβπ ) |
funcringcsetcALTV2.c | β’ πΆ = (Baseβπ) |
funcringcsetcALTV2.u | β’ (π β π β WUni) |
funcringcsetcALTV2.f | β’ (π β πΉ = (π₯ β π΅ β¦ (Baseβπ₯))) |
funcringcsetcALTV2.g | β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦)))) |
Ref | Expression |
---|---|
funcringcsetcALTV2lem5 | β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΊπ) = ( I βΎ (π RingHom π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcringcsetcALTV2.g | . . 3 β’ (π β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦)))) | |
2 | 1 | adantr 482 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅)) β πΊ = (π₯ β π΅, π¦ β π΅ β¦ ( I βΎ (π₯ RingHom π¦)))) |
3 | oveq12 7371 | . . . 4 β’ ((π₯ = π β§ π¦ = π) β (π₯ RingHom π¦) = (π RingHom π)) | |
4 | 3 | adantl 483 | . . 3 β’ (((π β§ (π β π΅ β§ π β π΅)) β§ (π₯ = π β§ π¦ = π)) β (π₯ RingHom π¦) = (π RingHom π)) |
5 | 4 | reseq2d 5942 | . 2 β’ (((π β§ (π β π΅ β§ π β π΅)) β§ (π₯ = π β§ π¦ = π)) β ( I βΎ (π₯ RingHom π¦)) = ( I βΎ (π RingHom π))) |
6 | simprl 770 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅)) β π β π΅) | |
7 | simprr 772 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅)) β π β π΅) | |
8 | ovexd 7397 | . . 3 β’ ((π β§ (π β π΅ β§ π β π΅)) β (π RingHom π) β V) | |
9 | 8 | resiexd 7171 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅)) β ( I βΎ (π RingHom π)) β V) |
10 | 2, 5, 6, 7, 9 | ovmpod 7512 | 1 β’ ((π β§ (π β π΅ β§ π β π΅)) β (ππΊπ) = ( I βΎ (π RingHom π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 β¦ cmpt 5193 I cid 5535 βΎ cres 5640 βcfv 6501 (class class class)co 7362 β cmpo 7364 WUnicwun 10643 Basecbs 17090 SetCatcsetc 17968 RingHom crh 20152 RingCatcringc 46375 |
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-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3066 df-rex 3075 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-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 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-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 |
This theorem is referenced by: funcringcsetcALTV2lem6 46413 funcringcsetcALTV2lem7 46414 funcringcsetcALTV2lem8 46415 funcringcsetcALTV2lem9 46416 |
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