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Mirrors > Home > ILE Home > Th. List > dmaddpi | GIF version |
Description: Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
dmaddpi | ⊢ dom +N = (N × N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 4835 | . . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o ) | |
2 | fnoa 6336 | . . . . 5 ⊢ +o Fn (On × On) | |
3 | fndm 5217 | . . . . 5 ⊢ ( +o Fn (On × On) → dom +o = (On × On)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ dom +o = (On × On) |
5 | 4 | ineq2i 3269 | . . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On)) |
6 | 1, 5 | eqtri 2158 | . 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On)) |
7 | df-pli 7106 | . . 3 ⊢ +N = ( +o ↾ (N × N)) | |
8 | 7 | dmeqi 4735 | . 2 ⊢ dom +N = dom ( +o ↾ (N × N)) |
9 | df-ni 7105 | . . . . . . 7 ⊢ N = (ω ∖ {∅}) | |
10 | difss 3197 | . . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω | |
11 | 9, 10 | eqsstri 3124 | . . . . . 6 ⊢ N ⊆ ω |
12 | omsson 4521 | . . . . . 6 ⊢ ω ⊆ On | |
13 | 11, 12 | sstri 3101 | . . . . 5 ⊢ N ⊆ On |
14 | anidm 393 | . . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On) | |
15 | 13, 14 | mpbir 145 | . . . 4 ⊢ (N ⊆ On ∧ N ⊆ On) |
16 | xpss12 4641 | . . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On)) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ (N × N) ⊆ (On × On) |
18 | dfss 3080 | . . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On))) | |
19 | 17, 18 | mpbi 144 | . 2 ⊢ (N × N) = ((N × N) ∩ (On × On)) |
20 | 6, 8, 19 | 3eqtr4i 2168 | 1 ⊢ dom +N = (N × N) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∖ cdif 3063 ∩ cin 3065 ⊆ wss 3066 ∅c0 3358 {csn 3522 Oncon0 4280 ωcom 4499 × cxp 4532 dom cdm 4534 ↾ cres 4536 Fn wfn 5113 +o coa 6303 Ncnpi 7073 +N cpli 7074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 df-ni 7105 df-pli 7106 |
This theorem is referenced by: (None) |
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