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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 8712 | . 2 ⊢ +no Fn (On × On) | |
2 | naddcl 8714 | . . . 4 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 +no 𝑧) ∈ On) | |
3 | 2 | rgen2 3197 | . . 3 ⊢ ∀𝑦 ∈ On ∀𝑧 ∈ On (𝑦 +no 𝑧) ∈ On |
4 | fveq2 6907 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ( +no ‘𝑥) = ( +no ‘〈𝑦, 𝑧〉)) | |
5 | df-ov 7434 | . . . . . 6 ⊢ (𝑦 +no 𝑧) = ( +no ‘〈𝑦, 𝑧〉) | |
6 | 4, 5 | eqtr4di 2793 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → ( +no ‘𝑥) = (𝑦 +no 𝑧)) |
7 | 6 | eleq1d 2824 | . . . 4 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (( +no ‘𝑥) ∈ On ↔ (𝑦 +no 𝑧) ∈ On)) |
8 | 7 | ralxp 5855 | . . 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 1537 ∈ wcel 2106 ∀wral 3059 〈cop 4637 × cxp 5687 Oncon0 6386 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 +no cnadd 8702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-frecs 8305 df-nadd 8703 |
This theorem is referenced by: naddunif 8730 naddasslem1 8731 naddasslem2 8732 |
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