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Mirrors > Home > ILE Home > Th. List > dmmulpi | GIF version |
Description: Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
dmmulpi | ⊢ dom ·N = (N × N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 4899 | . . 3 ⊢ dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o ) | |
2 | fnom 6409 | . . . . 5 ⊢ ·o Fn (On × On) | |
3 | fndm 5281 | . . . . 5 ⊢ ( ·o Fn (On × On) → dom ·o = (On × On)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ dom ·o = (On × On) |
5 | 4 | ineq2i 3315 | . . 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-mi 7238 | . . 3 ⊢ ·N = ( ·o ↾ (N × N)) | |
8 | 7 | dmeqi 4799 | . 2 ⊢ dom ·N = dom ( ·o ↾ (N × N)) |
9 | df-ni 7236 | . . . . . . 7 ⊢ N = (ω ∖ {∅}) | |
10 | difss 3243 | . . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω | |
11 | 9, 10 | eqsstri 3169 | . . . . . 6 ⊢ N ⊆ ω |
12 | omsson 4584 | . . . . . 6 ⊢ ω ⊆ On | |
13 | 11, 12 | sstri 3146 | . . . . 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 4705 | . . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On)) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ (N × N) ⊆ (On × On) |
18 | dfss 3125 | . . 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 3108 ∩ cin 3110 ⊆ wss 3111 ∅c0 3404 {csn 3570 Oncon0 4335 ωcom 4561 × cxp 4596 dom cdm 4598 ↾ cres 4600 Fn wfn 5177 ·o comu 6373 Ncnpi 7204 ·N cmi 7206 |
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 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
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 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-oadd 6379 df-omul 6380 df-ni 7236 df-mi 7238 |
This theorem is referenced by: (None) |
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