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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 8622 | . 2 β’ +no Fn (On Γ On) | |
2 | naddcl 8624 | . . . 4 β’ ((π¦ β On β§ π§ β On) β (π¦ +no π§) β On) | |
3 | 2 | rgen2 3195 | . . 3 β’ βπ¦ β On βπ§ β On (π¦ +no π§) β On |
4 | fveq2 6843 | . . . . . 6 β’ (π₯ = β¨π¦, π§β© β ( +no βπ₯) = ( +no ββ¨π¦, π§β©)) | |
5 | df-ov 7361 | . . . . . 6 β’ (π¦ +no π§) = ( +no ββ¨π¦, π§β©) | |
6 | 4, 5 | eqtr4di 2795 | . . . . 5 β’ (π₯ = β¨π¦, π§β© β ( +no βπ₯) = (π¦ +no π§)) |
7 | 6 | eleq1d 2823 | . . . 4 β’ (π₯ = β¨π¦, π§β© β (( +no βπ₯) β On β (π¦ +no π§) β On)) |
8 | 7 | ralxp 5798 | . . 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 7067 | . 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 3065 β¨cop 4593 Γ cxp 5632 Oncon0 6318 Fn wfn 6492 βΆwf 6493 βcfv 6497 (class class class)co 7358 +no cnadd 8612 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 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 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-frecs 8213 df-nadd 8613 |
This theorem is referenced by: naddunif 8638 naddasslem1 8639 naddasslem2 8640 |
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