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Mirrors > Home > MPE Home > Th. List > mulclsr | Structured version Visualization version GIF version |
Description: Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulclsr | ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 11080 | . . 3 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 7425 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → ([〈𝑥, 𝑦〉] ~R ·R [〈𝑧, 𝑤〉] ~R ) = (𝐴 ·R [〈𝑧, 𝑤〉] ~R )) | |
3 | 2 | eleq1d 2810 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → (([〈𝑥, 𝑦〉] ~R ·R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 ·R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ))) |
4 | oveq2 7426 | . . . 4 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → (𝐴 ·R [〈𝑧, 𝑤〉] ~R ) = (𝐴 ·R 𝐵)) | |
5 | 4 | eleq1d 2810 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ~R = 𝐵 → ((𝐴 ·R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R ) ↔ (𝐴 ·R 𝐵) ∈ ((P × P) / ~R ))) |
6 | mulsrpr 11100 | . . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([〈𝑥, 𝑦〉] ~R ·R [〈𝑧, 𝑤〉] ~R ) = [〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉] ~R ) | |
7 | mulclpr 11044 | . . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) ∈ P) | |
8 | mulclpr 11044 | . . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) ∈ P) | |
9 | addclpr 11042 | . . . . . . . 8 ⊢ (((𝑥 ·P 𝑧) ∈ P ∧ (𝑦 ·P 𝑤) ∈ P) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) | |
10 | 7, 8, 9 | syl2an 594 | . . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑧 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) |
11 | 10 | an4s 658 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P) |
12 | mulclpr 11044 | . . . . . . . 8 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 ·P 𝑤) ∈ P) | |
13 | mulclpr 11044 | . . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 ·P 𝑧) ∈ P) | |
14 | addclpr 11042 | . . . . . . . 8 ⊢ (((𝑥 ·P 𝑤) ∈ P ∧ (𝑦 ·P 𝑧) ∈ P) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) | |
15 | 12, 13, 14 | syl2an 594 | . . . . . . 7 ⊢ (((𝑥 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑦 ∈ P ∧ 𝑧 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) |
16 | 15 | an42s 659 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) |
17 | 11, 16 | jca 510 | . . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P)) |
18 | opelxpi 5715 | . . . . 5 ⊢ ((((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) ∈ P ∧ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) ∈ P) → 〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉 ∈ (P × P)) | |
19 | enrex 11091 | . . . . . 6 ⊢ ~R ∈ V | |
20 | 19 | ecelqsi 8791 | . . . . 5 ⊢ (〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉 ∈ (P × P) → [〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉] ~R ∈ ((P × P) / ~R )) |
21 | 17, 18, 20 | 3syl 18 | . . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → [〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉] ~R ∈ ((P × P) / ~R )) |
22 | 6, 21 | eqeltrd 2825 | . . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([〈𝑥, 𝑦〉] ~R ·R [〈𝑧, 𝑤〉] ~R ) ∈ ((P × P) / ~R )) |
23 | 1, 3, 5, 22 | 2ecoptocl 8826 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ ((P × P) / ~R )) |
24 | 23, 1 | eleqtrrdi 2836 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 〈cop 4636 × cxp 5676 (class class class)co 7418 [cec 8722 / cqs 8723 Pcnp 10883 +P cpp 10885 ·P cmp 10886 ~R cer 10888 Rcnr 10889 ·R cmr 10894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6306 df-ord 6373 df-on 6374 df-lim 6375 df-suc 6376 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 7421 df-oprab 7422 df-mpo 7423 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-oadd 8490 df-omul 8491 df-er 8724 df-ec 8726 df-qs 8730 df-ni 10896 df-pli 10897 df-mi 10898 df-lti 10899 df-plpq 10932 df-mpq 10933 df-ltpq 10934 df-enq 10935 df-nq 10936 df-erq 10937 df-plq 10938 df-mq 10939 df-1nq 10940 df-rq 10941 df-ltnq 10942 df-np 11005 df-plp 11007 df-mp 11008 df-ltp 11009 df-enr 11079 df-nr 11080 df-mr 11082 |
This theorem is referenced by: dmmulsr 11110 negexsr 11126 sqgt0sr 11130 recexsr 11131 map2psrpr 11134 mulresr 11163 axmulf 11170 axmulrcl 11178 axmulass 11181 axdistr 11182 axrnegex 11186 |
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