Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > axmulrcl | GIF version |
Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 7075 be used later. Instead, in most cases use remulcl 7101. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.) |
Ref | Expression |
---|---|
axmulrcl | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 6997 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 6997 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | oveq1 5539 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = (𝐴 · 〈𝑦, 0R〉)) | |
4 | 3 | eleq1d 2147 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 · 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 · 〈𝑦, 0R〉) ∈ ℝ)) |
5 | oveq2 5540 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 · 〈𝑦, 0R〉) = (𝐴 · 𝐵)) | |
6 | 5 | eleq1d 2147 | . 2 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 · 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 · 𝐵) ∈ ℝ)) |
7 | mulresr 7006 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = 〈(𝑥 ·R 𝑦), 0R〉) | |
8 | mulclsr 6931 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 ·R 𝑦) ∈ R) | |
9 | opelreal 6996 | . . . 4 ⊢ (〈(𝑥 ·R 𝑦), 0R〉 ∈ ℝ ↔ (𝑥 ·R 𝑦) ∈ R) | |
10 | 8, 9 | sylibr 132 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈(𝑥 ·R 𝑦), 0R〉 ∈ ℝ) |
11 | 7, 10 | eqeltrd 2155 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ∈ ℝ) |
12 | 1, 2, 4, 6, 11 | 2gencl 2632 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 〈cop 3401 (class class class)co 5532 Rcnr 6487 0Rc0r 6488 ·R cmr 6492 ℝcr 6980 · cmul 6986 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-iplp 6658 df-imp 6659 df-enr 6903 df-nr 6904 df-plr 6905 df-mr 6906 df-0r 6908 df-m1r 6910 df-c 6987 df-r 6991 df-mul 6993 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |