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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 5759 | . . 3 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ dom +o ) | |
2 | fnoa 7987 | . . . . 5 ⊢ +o Fn (On × On) | |
3 | fndm 6328 | . . . . 5 ⊢ ( +o Fn (On × On) → dom +o = (On × On)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ dom +o = (On × On) |
5 | 4 | ineq2i 4108 | . . 3 ⊢ ((N × N) ∩ dom +o ) = ((N × N) ∩ (On × On)) |
6 | 1, 5 | eqtri 2818 | . 2 ⊢ dom ( +o ↾ (N × N)) = ((N × N) ∩ (On × On)) |
7 | df-pli 10144 | . . 3 ⊢ +N = ( +o ↾ (N × N)) | |
8 | 7 | dmeqi 5662 | . 2 ⊢ dom +N = dom ( +o ↾ (N × N)) |
9 | df-ni 10143 | . . . . . . 7 ⊢ N = (ω ∖ {∅}) | |
10 | difss 4031 | . . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω | |
11 | 9, 10 | eqsstri 3924 | . . . . . 6 ⊢ N ⊆ ω |
12 | omsson 7443 | . . . . . 6 ⊢ ω ⊆ On | |
13 | 11, 12 | sstri 3900 | . . . . 5 ⊢ N ⊆ On |
14 | anidm 565 | . . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On) | |
15 | 13, 14 | mpbir 232 | . . . 4 ⊢ (N ⊆ On ∧ N ⊆ On) |
16 | xpss12 5461 | . . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On)) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ (N × N) ⊆ (On × On) |
18 | dfss 3877 | . . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On))) | |
19 | 17, 18 | mpbi 231 | . 2 ⊢ (N × N) = ((N × N) ∩ (On × On)) |
20 | 6, 8, 19 | 3eqtr4i 2828 | 1 ⊢ dom +N = (N × N) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1522 ∖ cdif 3858 ∩ cin 3860 ⊆ wss 3861 ∅c0 4213 {csn 4474 × cxp 5444 dom cdm 5446 ↾ cres 5448 Oncon0 6069 Fn wfn 6223 ωcom 7439 +o coa 7953 Ncnpi 10115 +N cpli 10116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-fv 6236 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-oadd 7960 df-ni 10143 df-pli 10144 |
This theorem is referenced by: addcompi 10165 addasspi 10166 distrpi 10169 addcanpi 10170 addnidpi 10172 ltapi 10174 |
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