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Mirrors > Home > MPE Home > Th. List > naddf | Structured version Visualization version GIF version |
Description: Function statement for natural addition. (Contributed by Scott Fenton, 20-Jan-2025.) |
Ref | Expression |
---|---|
naddf | β’ +no :(On Γ On)βΆOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | naddfn 8674 | . 2 β’ +no Fn (On Γ On) | |
2 | naddcl 8676 | . . . 4 β’ ((π¦ β On β§ π§ β On) β (π¦ +no π§) β On) | |
3 | 2 | rgen2 3198 | . . 3 β’ βπ¦ β On βπ§ β On (π¦ +no π§) β On |
4 | fveq2 6892 | . . . . . 6 β’ (π₯ = β¨π¦, π§β© β ( +no βπ₯) = ( +no ββ¨π¦, π§β©)) | |
5 | df-ov 7412 | . . . . . 6 β’ (π¦ +no π§) = ( +no ββ¨π¦, π§β©) | |
6 | 4, 5 | eqtr4di 2791 | . . . . 5 β’ (π₯ = β¨π¦, π§β© β ( +no βπ₯) = (π¦ +no π§)) |
7 | 6 | eleq1d 2819 | . . . 4 β’ (π₯ = β¨π¦, π§β© β (( +no βπ₯) β On β (π¦ +no π§) β On)) |
8 | 7 | ralxp 5842 | . . 3 β’ (βπ₯ β (On Γ On)( +no βπ₯) β On β βπ¦ β On βπ§ β On (π¦ +no π§) β On) |
9 | 3, 8 | mpbir 230 | . 2 β’ βπ₯ β (On Γ On)( +no βπ₯) β On |
10 | ffnfv 7118 | . 2 β’ ( +no :(On Γ On)βΆOn β ( +no Fn (On Γ On) β§ βπ₯ β (On Γ On)( +no βπ₯) β On)) | |
11 | 1, 9, 10 | mpbir2an 710 | 1 β’ +no :(On Γ On)βΆOn |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 βwral 3062 β¨cop 4635 Γ cxp 5675 Oncon0 6365 Fn wfn 6539 βΆwf 6540 βcfv 6544 (class class class)co 7409 +no cnadd 8664 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-frecs 8266 df-nadd 8665 |
This theorem is referenced by: naddunif 8692 naddasslem1 8693 naddasslem2 8694 |
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