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Mirrors > Home > MPE Home > Th. List > hsmexlem8 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 10430. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
Ref | Expression |
---|---|
hsmexlem7.h | β’ π» = (rec((π§ β V β¦ (harβπ« (π Γ π§))), (harβπ« π)) βΎ Ο) |
Ref | Expression |
---|---|
hsmexlem8 | β’ (π β Ο β (π»βsuc π) = (harβπ« (π Γ (π»βπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6904 | . 2 β’ (harβπ« (π Γ (π»βπ))) β V | |
2 | hsmexlem7.h | . . 3 β’ π» = (rec((π§ β V β¦ (harβπ« (π Γ π§))), (harβπ« π)) βΎ Ο) | |
3 | xpeq2 5697 | . . . . 5 β’ (π = π§ β (π Γ π) = (π Γ π§)) | |
4 | 3 | pweqd 4619 | . . . 4 β’ (π = π§ β π« (π Γ π) = π« (π Γ π§)) |
5 | 4 | fveq2d 6895 | . . 3 β’ (π = π§ β (harβπ« (π Γ π)) = (harβπ« (π Γ π§))) |
6 | xpeq2 5697 | . . . . 5 β’ (π = (π»βπ) β (π Γ π) = (π Γ (π»βπ))) | |
7 | 6 | pweqd 4619 | . . . 4 β’ (π = (π»βπ) β π« (π Γ π) = π« (π Γ (π»βπ))) |
8 | 7 | fveq2d 6895 | . . 3 β’ (π = (π»βπ) β (harβπ« (π Γ π)) = (harβπ« (π Γ (π»βπ)))) |
9 | 2, 5, 8 | frsucmpt2 8443 | . 2 β’ ((π β Ο β§ (harβπ« (π Γ (π»βπ))) β V) β (π»βsuc π) = (harβπ« (π Γ (π»βπ)))) |
10 | 1, 9 | mpan2 688 | 1 β’ (π β Ο β (π»βsuc π) = (harβπ« (π Γ (π»βπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 Vcvv 3473 π« cpw 4602 β¦ cmpt 5231 Γ cxp 5674 βΎ cres 5678 suc csuc 6366 βcfv 6543 Οcom 7858 reccrdg 8412 harchar 9554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7859 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 |
This theorem is referenced by: hsmexlem9 10423 hsmexlem4 10427 |
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