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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 10757 be used later. Instead, in most cases use remulcl 10779. (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 10710 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 10710 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | oveq1 7198 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = (𝐴 · 〈𝑦, 0R〉)) | |
4 | 3 | eleq1d 2815 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 · 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 · 〈𝑦, 0R〉) ∈ ℝ)) |
5 | oveq2 7199 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 · 〈𝑦, 0R〉) = (𝐴 · 𝐵)) | |
6 | 5 | eleq1d 2815 | . 2 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 · 〈𝑦, 0R〉) ∈ ℝ ↔ (𝐴 · 𝐵) ∈ ℝ)) |
7 | mulresr 10718 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) = 〈(𝑥 ·R 𝑦), 0R〉) | |
8 | mulclsr 10663 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 ·R 𝑦) ∈ R) | |
9 | opelreal 10709 | . . . 4 ⊢ (〈(𝑥 ·R 𝑦), 0R〉 ∈ ℝ ↔ (𝑥 ·R 𝑦) ∈ R) | |
10 | 8, 9 | sylibr 237 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → 〈(𝑥 ·R 𝑦), 0R〉 ∈ ℝ) |
11 | 7, 10 | eqeltrd 2831 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 · 〈𝑦, 0R〉) ∈ ℝ) |
12 | 1, 2, 4, 6, 11 | 2gencl 3438 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 〈cop 4533 (class class class)co 7191 Rcnr 10444 0Rc0r 10445 ·R cmr 10449 ℝcr 10693 · cmul 10699 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-omul 8185 df-er 8369 df-ec 8371 df-qs 8375 df-ni 10451 df-pli 10452 df-mi 10453 df-lti 10454 df-plpq 10487 df-mpq 10488 df-ltpq 10489 df-enq 10490 df-nq 10491 df-erq 10492 df-plq 10493 df-mq 10494 df-1nq 10495 df-rq 10496 df-ltnq 10497 df-np 10560 df-1p 10561 df-plp 10562 df-mp 10563 df-ltp 10564 df-enr 10634 df-nr 10635 df-plr 10636 df-mr 10637 df-0r 10639 df-m1r 10641 df-c 10700 df-r 10704 df-mul 10706 |
This theorem is referenced by: (None) |
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