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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 4902 | . . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o ) | |
2 | fnoa 6409 | . . . . 5 ⊢ +o Fn (On × On) | |
3 | fndm 5284 | . . . . 5 ⊢ ( +o Fn (On × On) → dom +o = (On × On)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ dom +o = (On × On) |
5 | 4 | ineq2i 3318 | . . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On)) |
6 | 1, 5 | eqtri 2185 | . 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On)) |
7 | df-pli 7240 | . . 3 ⊢ +N = ( +o ↾ (N × N)) | |
8 | 7 | dmeqi 4802 | . 2 ⊢ dom +N = dom ( +o ↾ (N × N)) |
9 | df-ni 7239 | . . . . . . 7 ⊢ N = (ω ∖ {∅}) | |
10 | difss 3246 | . . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω | |
11 | 9, 10 | eqsstri 3172 | . . . . . 6 ⊢ N ⊆ ω |
12 | omsson 4587 | . . . . . 6 ⊢ ω ⊆ On | |
13 | 11, 12 | sstri 3149 | . . . . 5 ⊢ N ⊆ On |
14 | anidm 394 | . . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On) | |
15 | 13, 14 | mpbir 145 | . . . 4 ⊢ (N ⊆ On ∧ N ⊆ On) |
16 | xpss12 4708 | . . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On)) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ (N × N) ⊆ (On × On) |
18 | dfss 3128 | . . 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 2195 | 1 ⊢ dom +N = (N × N) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∖ cdif 3111 ∩ cin 3113 ⊆ wss 3114 ∅c0 3407 {csn 3573 Oncon0 4338 ωcom 4564 × cxp 4599 dom cdm 4601 ↾ cres 4603 Fn wfn 5180 +o coa 6375 Ncnpi 7207 +N cpli 7208 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-iord 4341 df-on 4343 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-recs 6267 df-irdg 6332 df-oadd 6382 df-ni 7239 df-pli 7240 |
This theorem is referenced by: (None) |
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