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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 | ⊢ +no :(On × On)⟶On | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | naddfn 8713 | . 2 ⊢ +no Fn (On × On) | |
| 2 | naddcl 8715 | . . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 +no 𝑧) ∈ On) | |
| 3 | 2 | rgen2 3199 | . . 3 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On | 
| 4 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ( +no ‘𝑥) = ( +no ‘〈𝑦, 𝑧〉)) | |
| 5 | df-ov 7434 | . . . . . 6 ⊢ (𝑦 +no 𝑧) = ( +no ‘〈𝑦, 𝑧〉) | |
| 6 | 4, 5 | eqtr4di 2795 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ( +no ‘𝑥) = (𝑦 +no 𝑧)) | 
| 7 | 6 | eleq1d 2826 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (( +no ‘𝑥) ∈ On ↔ (𝑦 +no 𝑧) ∈ On)) | 
| 8 | 7 | ralxp 5852 | . . 3 ⊢ (∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On ↔ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On) | 
| 9 | 3, 8 | mpbir 231 | . 2 ⊢ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On | 
| 10 | ffnfv 7139 | . 2 ⊢ ( +no :(On × On)⟶On ↔ ( +no Fn (On × On) ∧ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On)) | |
| 11 | 1, 9, 10 | mpbir2an 711 | 1 ⊢ +no :(On × On)⟶On | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 〈cop 4632 × cxp 5683 Oncon0 6384 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 +no cnadd 8703 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-frecs 8306 df-nadd 8704 | 
| This theorem is referenced by: naddunif 8731 naddasslem1 8732 naddasslem2 8733 | 
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