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Mirrors > Home > MPE Home > Th. List > mulnqf | Structured version Visualization version GIF version |
Description: Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulnqf | ⊢ ·Q :(Q × Q)⟶Q |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqerf 10954 | . . . 4 ⊢ [Q]:(N × N)⟶Q | |
2 | mulpqf 10970 | . . . 4 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | |
3 | fco 6747 | . . . 4 ⊢ (([Q]:(N × N)⟶Q ∧ ·pQ :((N × N) × (N × N))⟶(N × N)) → ([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q) | |
4 | 1, 2, 3 | mp2an 691 | . . 3 ⊢ ([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q |
5 | elpqn 10949 | . . . . 5 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
6 | 5 | ssriv 3984 | . . . 4 ⊢ Q ⊆ (N × N) |
7 | xpss12 5693 | . . . 4 ⊢ ((Q ⊆ (N × N) ∧ Q ⊆ (N × N)) → (Q × Q) ⊆ ((N × N) × (N × N))) | |
8 | 6, 6, 7 | mp2an 691 | . . 3 ⊢ (Q × Q) ⊆ ((N × N) × (N × N)) |
9 | fssres 6763 | . . 3 ⊢ ((([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q ∧ (Q × Q) ⊆ ((N × N) × (N × N))) → (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q) | |
10 | 4, 8, 9 | mp2an 691 | . 2 ⊢ (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q |
11 | df-mq 10939 | . . 3 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q)) | |
12 | 11 | feq1i 6713 | . 2 ⊢ ( ·Q :(Q × Q)⟶Q ↔ (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q) |
13 | 10, 12 | mpbir 230 | 1 ⊢ ·Q :(Q × Q)⟶Q |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3947 × cxp 5676 ↾ cres 5680 ∘ ccom 5682 ⟶wf 6544 Ncnpi 10868 ·pQ cmpq 10873 Qcnq 10876 [Q]cerq 10878 ·Q cmq 10880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ni 10896 df-mi 10898 df-lti 10899 df-mpq 10933 df-enq 10935 df-nq 10936 df-erq 10937 df-mq 10939 df-1nq 10940 |
This theorem is referenced by: mulcomnq 10977 mulerpq 10981 mulassnq 10983 distrnq 10985 recmulnq 10988 recclnq 10990 dmrecnq 10992 ltmnq 10996 prlem936 11071 |
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