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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 4810 | . . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o ) | |
2 | fnoa 6311 | . . . . 5 ⊢ +o Fn (On × On) | |
3 | fndm 5192 | . . . . 5 ⊢ ( +o Fn (On × On) → dom +o = (On × On)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ dom +o = (On × On) |
5 | 4 | ineq2i 3244 | . . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On)) |
6 | 1, 5 | eqtri 2138 | . 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On)) |
7 | df-pli 7081 | . . 3 ⊢ +N = ( +o ↾ (N × N)) | |
8 | 7 | dmeqi 4710 | . 2 ⊢ dom +N = dom ( +o ↾ (N × N)) |
9 | df-ni 7080 | . . . . . . 7 ⊢ N = (ω ∖ {∅}) | |
10 | difss 3172 | . . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω | |
11 | 9, 10 | eqsstri 3099 | . . . . . 6 ⊢ N ⊆ ω |
12 | omsson 4496 | . . . . . 6 ⊢ ω ⊆ On | |
13 | 11, 12 | sstri 3076 | . . . . 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 4616 | . . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On)) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ (N × N) ⊆ (On × On) |
18 | dfss 3055 | . . 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 2148 | 1 ⊢ dom +N = (N × N) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∖ cdif 3038 ∩ cin 3040 ⊆ wss 3041 ∅c0 3333 {csn 3497 Oncon0 4255 ωcom 4474 × cxp 4507 dom cdm 4509 ↾ cres 4511 Fn wfn 5088 +o coa 6278 Ncnpi 7048 +N cpli 7049 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-oadd 6285 df-ni 7080 df-pli 7081 |
This theorem is referenced by: (None) |
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