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Mirrors > Home > MPE Home > Th. List > Mathboxes > funcringcsetcALTV2lem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for funcringcsetcALTV2 47284. (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 480 | . 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 7423 | . . . 4 β’ ((π₯ = π β§ π¦ = π) β (π₯ RingHom π¦) = (π RingHom π)) | |
4 | 3 | adantl 481 | . . 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 5979 | . 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 7449 | . . 3 β’ ((π β§ (π β π΅ β§ π β π΅)) β (π RingHom π) β V) | |
9 | 8 | resiexd 7222 | . 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 7567 | 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 395 = wceq 1534 β wcel 2099 Vcvv 3469 β¦ cmpt 5225 I cid 5569 βΎ cres 5674 βcfv 6542 (class class class)co 7414 β cmpo 7416 WUnicwun 10715 Basecbs 17171 SetCatcsetc 18055 RingHom crh 20397 RingCatcringc 20567 |
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-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 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-ral 3057 df-rex 3066 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-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 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-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 |
This theorem is referenced by: funcringcsetcALTV2lem6 47280 funcringcsetcALTV2lem7 47281 funcringcsetcALTV2lem8 47282 funcringcsetcALTV2lem9 47283 |
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