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Mirrors > Home > MPE Home > Th. List > dmmulpi | Structured version Visualization version GIF version |
Description: Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmmulpi | ⊢ dom ·N = (N × N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 5339 | . . 3 ⊢ dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ dom ·𝑜 ) | |
2 | fnom 7476 | . . . . 5 ⊢ ·𝑜 Fn (On × On) | |
3 | fndm 5904 | . . . . 5 ⊢ ( ·𝑜 Fn (On × On) → dom ·𝑜 = (On × On)) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ dom ·𝑜 = (On × On) |
5 | 4 | ineq2i 3773 | . . 3 ⊢ ((N × N) ∩ dom ·𝑜 ) = ((N × N) ∩ (On × On)) |
6 | 1, 5 | eqtri 2632 | . 2 ⊢ dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On)) |
7 | df-mi 9575 | . . 3 ⊢ ·N = ( ·𝑜 ↾ (N × N)) | |
8 | 7 | dmeqi 5247 | . 2 ⊢ dom ·N = dom ( ·𝑜 ↾ (N × N)) |
9 | df-ni 9573 | . . . . . . 7 ⊢ N = (ω ∖ {∅}) | |
10 | difss 3699 | . . . . . . 7 ⊢ (ω ∖ {∅}) ⊆ ω | |
11 | 9, 10 | eqsstri 3598 | . . . . . 6 ⊢ N ⊆ ω |
12 | omsson 6961 | . . . . . 6 ⊢ ω ⊆ On | |
13 | 11, 12 | sstri 3577 | . . . . 5 ⊢ N ⊆ On |
14 | anidm 674 | . . . . 5 ⊢ ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On) | |
15 | 13, 14 | mpbir 220 | . . . 4 ⊢ (N ⊆ On ∧ N ⊆ On) |
16 | xpss12 5148 | . . . 4 ⊢ ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On)) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ (N × N) ⊆ (On × On) |
18 | dfss 3555 | . . 3 ⊢ ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On))) | |
19 | 17, 18 | mpbi 219 | . 2 ⊢ (N × N) = ((N × N) ∩ (On × On)) |
20 | 6, 8, 19 | 3eqtr4i 2642 | 1 ⊢ dom ·N = (N × N) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 × cxp 5036 dom cdm 5038 ↾ cres 5040 Oncon0 5640 Fn wfn 5799 ωcom 6957 ·𝑜 comu 7445 Ncnpi 9545 ·N cmi 9547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-omul 7452 df-ni 9573 df-mi 9575 |
This theorem is referenced by: mulcompi 9597 mulasspi 9598 distrpi 9599 mulcanpi 9601 ltmpi 9605 ordpipq 9643 |
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