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Mirrors > Home > ILE Home > Th. List > irrmulap | GIF version |
Description: The product of an irrational with a nonzero rational is irrational. By irrational we mean apart from any rational number. For a similar theorem with not rational in place of irrational, see irrmul 9698. (Contributed by Jim Kingdon, 25-Aug-2025.) |
Ref | Expression |
---|---|
irrmulap.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
irrmulap.aq | ⊢ (𝜑 → ∀𝑞 ∈ ℚ 𝐴 # 𝑞) |
irrmulap.b | ⊢ (𝜑 → 𝐵 ∈ ℚ) |
irrmulap.b0 | ⊢ (𝜑 → 𝐵 ≠ 0) |
irrmulap.q | ⊢ (𝜑 → 𝑄 ∈ ℚ) |
Ref | Expression |
---|---|
irrmulap | ⊢ (𝜑 → (𝐴 · 𝐵) # 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4029 | . . . 4 ⊢ (𝑞 = (𝑄 / 𝐵) → (𝐴 # 𝑞 ↔ 𝐴 # (𝑄 / 𝐵))) | |
2 | irrmulap.aq | . . . 4 ⊢ (𝜑 → ∀𝑞 ∈ ℚ 𝐴 # 𝑞) | |
3 | irrmulap.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℚ) | |
4 | irrmulap.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℚ) | |
5 | irrmulap.b0 | . . . . 5 ⊢ (𝜑 → 𝐵 ≠ 0) | |
6 | qdivcl 9694 | . . . . 5 ⊢ ((𝑄 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝑄 / 𝐵) ∈ ℚ) | |
7 | 3, 4, 5, 6 | syl3anc 1249 | . . . 4 ⊢ (𝜑 → (𝑄 / 𝐵) ∈ ℚ) |
8 | 1, 2, 7 | rspcdva 2865 | . . 3 ⊢ (𝜑 → 𝐴 # (𝑄 / 𝐵)) |
9 | qcn 9685 | . . . . 5 ⊢ ((𝑄 / 𝐵) ∈ ℚ → (𝑄 / 𝐵) ∈ ℂ) | |
10 | 7, 9 | syl 14 | . . . 4 ⊢ (𝜑 → (𝑄 / 𝐵) ∈ ℂ) |
11 | irrmulap.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
12 | 11 | recnd 8034 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | apsym 8611 | . . . 4 ⊢ (((𝑄 / 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝑄 / 𝐵) # 𝐴 ↔ 𝐴 # (𝑄 / 𝐵))) | |
14 | 10, 12, 13 | syl2anc 411 | . . 3 ⊢ (𝜑 → ((𝑄 / 𝐵) # 𝐴 ↔ 𝐴 # (𝑄 / 𝐵))) |
15 | 8, 14 | mpbird 167 | . 2 ⊢ (𝜑 → (𝑄 / 𝐵) # 𝐴) |
16 | qcn 9685 | . . . . 5 ⊢ (𝑄 ∈ ℚ → 𝑄 ∈ ℂ) | |
17 | 3, 16 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
18 | qcn 9685 | . . . . 5 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
19 | 4, 18 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
20 | 0z 9314 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
21 | zq 9677 | . . . . . . 7 ⊢ (0 ∈ ℤ → 0 ∈ ℚ) | |
22 | 20, 21 | ax-mp 5 | . . . . . 6 ⊢ 0 ∈ ℚ |
23 | qapne 9690 | . . . . . 6 ⊢ ((𝐵 ∈ ℚ ∧ 0 ∈ ℚ) → (𝐵 # 0 ↔ 𝐵 ≠ 0)) | |
24 | 4, 22, 23 | sylancl 413 | . . . . 5 ⊢ (𝜑 → (𝐵 # 0 ↔ 𝐵 ≠ 0)) |
25 | 5, 24 | mpbird 167 | . . . 4 ⊢ (𝜑 → 𝐵 # 0) |
26 | 17, 19, 12, 25 | apdivmuld 8818 | . . 3 ⊢ (𝜑 → ((𝑄 / 𝐵) # 𝐴 ↔ (𝐵 · 𝐴) # 𝑄)) |
27 | 19, 12 | mulcomd 8027 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝐴) = (𝐴 · 𝐵)) |
28 | 27 | breq1d 4035 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝐴) # 𝑄 ↔ (𝐴 · 𝐵) # 𝑄)) |
29 | 26, 28 | bitrd 188 | . 2 ⊢ (𝜑 → ((𝑄 / 𝐵) # 𝐴 ↔ (𝐴 · 𝐵) # 𝑄)) |
30 | 15, 29 | mpbid 147 | 1 ⊢ (𝜑 → (𝐴 · 𝐵) # 𝑄) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2160 ≠ wne 2360 ∀wral 2468 class class class wbr 4025 (class class class)co 5906 ℂcc 7856 ℝcr 7857 0cc0 7858 · cmul 7863 # cap 8586 / cdiv 8677 ℤcz 9303 ℚcq 9670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-mulrcl 7957 ax-addcom 7958 ax-mulcom 7959 ax-addass 7960 ax-mulass 7961 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-1rid 7965 ax-0id 7966 ax-rnegex 7967 ax-precex 7968 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-apti 7973 ax-pre-ltadd 7974 ax-pre-mulgt0 7975 ax-pre-mulext 7976 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-id 4318 df-po 4321 df-iso 4322 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-1st 6180 df-2nd 6181 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-reap 8580 df-ap 8587 df-div 8678 df-inn 8969 df-n0 9227 df-z 9304 df-q 9671 |
This theorem is referenced by: (None) |
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