![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > addsf | Structured version Visualization version GIF version |
Description: Function statement for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) |
Ref | Expression |
---|---|
addsf | ⊢ +s :( No × No )⟶ No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addsfn 27818 | . 2 ⊢ +s Fn ( No × No ) | |
2 | addscl 27838 | . . . 4 ⊢ ((𝑦 ∈ No ∧ 𝑧 ∈ No ) → (𝑦 +s 𝑧) ∈ No ) | |
3 | 2 | rgen2 3189 | . . 3 ⊢ ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑦 +s 𝑧) ∈ No |
4 | fveq2 6882 | . . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ( +s ‘𝑥) = ( +s ‘⟨𝑦, 𝑧⟩)) | |
5 | df-ov 7405 | . . . . . 6 ⊢ (𝑦 +s 𝑧) = ( +s ‘⟨𝑦, 𝑧⟩) | |
6 | 4, 5 | eqtr4di 2782 | . . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → ( +s ‘𝑥) = (𝑦 +s 𝑧)) |
7 | 6 | eleq1d 2810 | . . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (( +s ‘𝑥) ∈ No ↔ (𝑦 +s 𝑧) ∈ No )) |
8 | 7 | ralxp 5832 | . . 3 ⊢ (∀𝑥 ∈ ( No × No )( +s ‘𝑥) ∈ No ↔ ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑦 +s 𝑧) ∈ No ) |
9 | 3, 8 | mpbir 230 | . 2 ⊢ ∀𝑥 ∈ ( No × No )( +s ‘𝑥) ∈ No |
10 | ffnfv 7111 | . 2 ⊢ ( +s :( No × No )⟶ No ↔ ( +s Fn ( No × No ) ∧ ∀𝑥 ∈ ( No × No )( +s ‘𝑥) ∈ No )) | |
11 | 1, 9, 10 | mpbir2an 708 | 1 ⊢ +s :( No × No )⟶ No |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∀wral 3053 ⟨cop 4627 × cxp 5665 Fn wfn 6529 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 No csur 27513 +s cadds 27816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-1o 8462 df-2o 8463 df-nadd 8662 df-no 27516 df-slt 27517 df-bday 27518 df-sslt 27654 df-scut 27656 df-0s 27697 df-made 27714 df-old 27715 df-left 27717 df-right 27718 df-norec2 27806 df-adds 27817 |
This theorem is referenced by: addsfo 27840 |
Copyright terms: Public domain | W3C validator |