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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 8656 | . 2 ⊢ +no Fn (On × On) | |
2 | naddcl 8658 | . . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 +no 𝑧) ∈ On) | |
3 | 2 | rgen2 3196 | . . 3 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On |
4 | fveq2 6877 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ( +no ‘𝑥) = ( +no ‘〈𝑦, 𝑧〉)) | |
5 | df-ov 7395 | . . . . . 6 ⊢ (𝑦 +no 𝑧) = ( +no ‘〈𝑦, 𝑧〉) | |
6 | 4, 5 | eqtr4di 2789 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ( +no ‘𝑥) = (𝑦 +no 𝑧)) |
7 | 6 | eleq1d 2817 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (( +no ‘𝑥) ∈ On ↔ (𝑦 +no 𝑧) ∈ On)) |
8 | 7 | ralxp 5832 | . . 3 ⊢ (∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On ↔ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On) |
9 | 3, 8 | mpbir 230 | . 2 ⊢ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On |
10 | ffnfv 7101 | . 2 ⊢ ( +no :(On × On)⟶On ↔ ( +no Fn (On × On) ∧ ∀𝑥 ∈ (On × On)( +no ‘𝑥) ∈ On)) | |
11 | 1, 9, 10 | mpbir2an 709 | 1 ⊢ +no :(On × On)⟶On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ∀wral 3060 〈cop 4627 × cxp 5666 Oncon0 6352 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7392 +no cnadd 8646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5277 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4943 df-iun 4991 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-se 5624 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-pred 6288 df-ord 6355 df-on 6356 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7956 df-2nd 7957 df-frecs 8247 df-nadd 8647 |
This theorem is referenced by: naddunif 8674 naddasslem1 8675 naddasslem2 8676 |
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