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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 10544 | . . . 4 ⊢ [Q]:(N × N)⟶Q | |
2 | mulpqf 10560 | . . . 4 ⊢ ·pQ :((N × N) × (N × N))⟶(N × N) | |
3 | fco 6569 | . . . 4 ⊢ (([Q]:(N × N)⟶Q ∧ ·pQ :((N × N) × (N × N))⟶(N × N)) → ([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q) | |
4 | 1, 2, 3 | mp2an 692 | . . 3 ⊢ ([Q] ∘ ·pQ ):((N × N) × (N × N))⟶Q |
5 | elpqn 10539 | . . . . 5 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
6 | 5 | ssriv 3905 | . . . 4 ⊢ Q ⊆ (N × N) |
7 | xpss12 5566 | . . . 4 ⊢ ((Q ⊆ (N × N) ∧ Q ⊆ (N × N)) → (Q × Q) ⊆ ((N × N) × (N × N))) | |
8 | 6, 6, 7 | mp2an 692 | . . 3 ⊢ (Q × Q) ⊆ ((N × N) × (N × N)) |
9 | fssres 6585 | . . 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 692 | . 2 ⊢ (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q |
11 | df-mq 10529 | . . 3 ⊢ ·Q = (([Q] ∘ ·pQ ) ↾ (Q × Q)) | |
12 | 11 | feq1i 6536 | . 2 ⊢ ( ·Q :(Q × Q)⟶Q ↔ (([Q] ∘ ·pQ ) ↾ (Q × Q)):(Q × Q)⟶Q) |
13 | 10, 12 | mpbir 234 | 1 ⊢ ·Q :(Q × Q)⟶Q |
Colors of variables: wff setvar class |
Syntax hints: ⊆ wss 3866 × cxp 5549 ↾ cres 5553 ∘ ccom 5555 ⟶wf 6376 Ncnpi 10458 ·pQ cmpq 10463 Qcnq 10466 [Q]cerq 10468 ·Q cmq 10470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-omul 8207 df-er 8391 df-ni 10486 df-mi 10488 df-lti 10489 df-mpq 10523 df-enq 10525 df-nq 10526 df-erq 10527 df-mq 10529 df-1nq 10530 |
This theorem is referenced by: mulcomnq 10567 mulerpq 10571 mulassnq 10573 distrnq 10575 recmulnq 10578 recclnq 10580 dmrecnq 10582 ltmnq 10586 prlem936 10661 |
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