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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 9712. (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 4033 | . . . 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 9708 | . . . . 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 2869 | . . 3 ⊢ (𝜑 → 𝐴 # (𝑄 / 𝐵)) |
9 | qcn 9699 | . . . . 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 8048 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | apsym 8625 | . . . 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 9699 | . . . . 5 ⊢ (𝑄 ∈ ℚ → 𝑄 ∈ ℂ) | |
17 | 3, 16 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ ℂ) |
18 | qcn 9699 | . . . . 5 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
19 | 4, 18 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
20 | 0z 9328 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
21 | zq 9691 | . . . . . . 7 ⊢ (0 ∈ ℤ → 0 ∈ ℚ) | |
22 | 20, 21 | ax-mp 5 | . . . . . 6 ⊢ 0 ∈ ℚ |
23 | qapne 9704 | . . . . . 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 8832 | . . 3 ⊢ (𝜑 → ((𝑄 / 𝐵) # 𝐴 ↔ (𝐵 · 𝐴) # 𝑄)) |
27 | 19, 12 | mulcomd 8041 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝐴) = (𝐴 · 𝐵)) |
28 | 27 | breq1d 4039 | . . 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 2164 ≠ wne 2364 ∀wral 2472 class class class wbr 4029 (class class class)co 5918 ℂcc 7870 ℝcr 7871 0cc0 7872 · cmul 7877 # cap 8600 / cdiv 8691 ℤcz 9317 ℚcq 9684 |
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 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 |
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 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-po 4327 df-iso 4328 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-n0 9241 df-z 9318 df-q 9685 |
This theorem is referenced by: (None) |
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