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Mirrors > Home > MPE Home > Th. List > axmulrcl | Structured version Visualization version GIF version |
Description: Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 10594 be used later. Instead, in most cases use remulcl 10616. (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 10547 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 10547 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | oveq1 7157 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = (𝐴 · 〈𝑦, 0R〉)) | |
4 | 3 | eleq1d 2897 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 · 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 · 〈𝑦, 0R〉) ∈ ℝ)) |
5 | oveq2 7158 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 · 〈𝑦, 0R〉) = (𝐴 · 𝐵)) | |
6 | 5 | eleq1d 2897 | . 2 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 · 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 · 𝐵) ∈ ℝ)) |
7 | mulresr 10555 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = 〈(𝑥 ·R 𝑦), 0R〉) | |
8 | mulclsr 10500 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 ·R 𝑦) ∈ R) | |
9 | opelreal 10546 | . . . 4 ⊢ (〈(𝑥 ·R 𝑦), 0R〉 ∈ ℝ ↔ (𝑥 ·R 𝑦) ∈ R) | |
10 | 8, 9 | sylibr 236 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈(𝑥 ·R 𝑦), 0R〉 ∈ ℝ) |
11 | 7, 10 | eqeltrd 2913 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ∈ ℝ) |
12 | 1, 2, 4, 6, 11 | 2gencl 3536 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4567 (class class class)co 7150 Rcnr 10281 0Rc0r 10282 ·R cmr 10286 ℝcr 10530 · cmul 10536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-rq 10333 df-ltnq 10334 df-np 10397 df-1p 10398 df-plp 10399 df-mp 10400 df-ltp 10401 df-enr 10471 df-nr 10472 df-plr 10473 df-mr 10474 df-0r 10476 df-m1r 10478 df-c 10537 df-r 10541 df-mul 10543 |
This theorem is referenced by: (None) |
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