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| Mirrors > Home > MPE Home > Th. List > dmaddpi | Structured version Visualization version GIF version | ||
| Description: Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| dmaddpi | ⊢ dom +N = (N × N) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmres 6030 | . . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o ) | |
| 2 | fnoa 8546 | . . . . 5 ⊢ +o Fn (On × On) | |
| 3 | 2 | fndmi 6672 | . . . 4 ⊢ dom +o = (On × On) |
| 4 | 3 | ineq2i 4217 | . . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On)) |
| 5 | 1, 4 | eqtri 2765 | . 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On)) |
| 6 | df-pli 10913 | . . 3 ⊢ +N = ( +o ↾ (N × N)) | |
| 7 | 6 | dmeqi 5915 | . 2 ⊢ dom +N = dom ( +o ↾ (N × N)) |
| 8 | df-ni 10912 | . . . . . . 7 ⊢ N = (ω ∖ {∅}) | |
| 9 | difss 4136 | . . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω | |
| 10 | 8, 9 | eqsstri 4030 | . . . . . 6 ⊢ N ⊆ ω |
| 11 | omsson 7891 | . . . . . 6 ⊢ ω ⊆ On | |
| 12 | 10, 11 | sstri 3993 | . . . . 5 ⊢ N ⊆ On |
| 13 | anidm 564 | . . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On) | |
| 14 | 12, 13 | mpbir 231 | . . . 4 ⊢ (N ⊆ On ∧ N ⊆ On) |
| 15 | xpss12 5700 | . . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On)) | |
| 16 | 14, 15 | ax-mp 5 | . . 3 ⊢ (N × N) ⊆ (On × On) |
| 17 | dfss 3970 | . . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On))) | |
| 18 | 16, 17 | mpbi 230 | . 2 ⊢ (N × N) = ((N × N) ∩ (On × On)) |
| 19 | 5, 7, 18 | 3eqtr4i 2775 | 1 ⊢ dom +N = (N × N) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 × cxp 5683 dom cdm 5685 ↾ cres 5687 Oncon0 6384 ωcom 7887 +o coa 8503 Ncnpi 10884 +N cpli 10885 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-oadd 8510 df-ni 10912 df-pli 10913 |
| This theorem is referenced by: addcompi 10934 addasspi 10935 distrpi 10938 addcanpi 10939 addnidpi 10941 ltapi 10943 |
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