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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem36 | Structured version Visualization version GIF version |
Description: 𝐹 is an isomorphism. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fourierdlem36.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fourierdlem36.assr | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
fourierdlem36.f | ⊢ 𝐹 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴)) |
fourierdlem36.n | ⊢ 𝑁 = ((♯‘𝐴) − 1) |
Ref | Expression |
---|---|
fourierdlem36 | ⊢ (𝜑 → 𝐹 Isom < , < ((0...𝑁), 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fourierdlem36.f | . . 3 ⊢ 𝐹 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴)) | |
2 | fourierdlem36.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
3 | fourierdlem36.assr | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
4 | ltso 11240 | . . . . . . 7 ⊢ < Or ℝ | |
5 | soss 5566 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴)) | |
6 | 3, 4, 5 | mpisyl 21 | . . . . . 6 ⊢ (𝜑 → < Or 𝐴) |
7 | 0zd 12516 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℤ) | |
8 | eqid 2733 | . . . . . 6 ⊢ ((♯‘𝐴) + (0 − 1)) = ((♯‘𝐴) + (0 − 1)) | |
9 | 2, 6, 7, 8 | fzisoeu 43621 | . . . . 5 ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((0...((♯‘𝐴) + (0 − 1))), 𝐴)) |
10 | hashcl 14262 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
11 | 2, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
12 | 11 | nn0cnd 12480 | . . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ) |
13 | 1cnd 11155 | . . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ) | |
14 | 12, 13 | negsubd 11523 | . . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝐴) + -1) = ((♯‘𝐴) − 1)) |
15 | df-neg 11393 | . . . . . . . . . . 11 ⊢ -1 = (0 − 1) | |
16 | 15 | eqcomi 2742 | . . . . . . . . . 10 ⊢ (0 − 1) = -1 |
17 | 16 | oveq2i 7369 | . . . . . . . . 9 ⊢ ((♯‘𝐴) + (0 − 1)) = ((♯‘𝐴) + -1) |
18 | fourierdlem36.n | . . . . . . . . 9 ⊢ 𝑁 = ((♯‘𝐴) − 1) | |
19 | 14, 17, 18 | 3eqtr4g 2798 | . . . . . . . 8 ⊢ (𝜑 → ((♯‘𝐴) + (0 − 1)) = 𝑁) |
20 | 19 | oveq2d 7374 | . . . . . . 7 ⊢ (𝜑 → (0...((♯‘𝐴) + (0 − 1))) = (0...𝑁)) |
21 | isoeq4 7266 | . . . . . . 7 ⊢ ((0...((♯‘𝐴) + (0 − 1))) = (0...𝑁) → (𝑓 Isom < , < ((0...((♯‘𝐴) + (0 − 1))), 𝐴) ↔ 𝑓 Isom < , < ((0...𝑁), 𝐴))) | |
22 | 20, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑓 Isom < , < ((0...((♯‘𝐴) + (0 − 1))), 𝐴) ↔ 𝑓 Isom < , < ((0...𝑁), 𝐴))) |
23 | 22 | eubidv 2581 | . . . . 5 ⊢ (𝜑 → (∃!𝑓 𝑓 Isom < , < ((0...((♯‘𝐴) + (0 − 1))), 𝐴) ↔ ∃!𝑓 𝑓 Isom < , < ((0...𝑁), 𝐴))) |
24 | 9, 23 | mpbid 231 | . . . 4 ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((0...𝑁), 𝐴)) |
25 | iotacl 6483 | . . . 4 ⊢ (∃!𝑓 𝑓 Isom < , < ((0...𝑁), 𝐴) → (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴)) ∈ {𝑓 ∣ 𝑓 Isom < , < ((0...𝑁), 𝐴)}) | |
26 | 24, 25 | syl 17 | . . 3 ⊢ (𝜑 → (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴)) ∈ {𝑓 ∣ 𝑓 Isom < , < ((0...𝑁), 𝐴)}) |
27 | 1, 26 | eqeltrid 2838 | . 2 ⊢ (𝜑 → 𝐹 ∈ {𝑓 ∣ 𝑓 Isom < , < ((0...𝑁), 𝐴)}) |
28 | iotaex 6470 | . . . 4 ⊢ (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴)) ∈ V | |
29 | 1, 28 | eqeltri 2830 | . . 3 ⊢ 𝐹 ∈ V |
30 | isoeq1 7263 | . . 3 ⊢ (𝑓 = 𝐹 → (𝑓 Isom < , < ((0...𝑁), 𝐴) ↔ 𝐹 Isom < , < ((0...𝑁), 𝐴))) | |
31 | 29, 30 | elab 3631 | . 2 ⊢ (𝐹 ∈ {𝑓 ∣ 𝑓 Isom < , < ((0...𝑁), 𝐴)} ↔ 𝐹 Isom < , < ((0...𝑁), 𝐴)) |
32 | 27, 31 | sylib 217 | 1 ⊢ (𝜑 → 𝐹 Isom < , < ((0...𝑁), 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∃!weu 2563 {cab 2710 Vcvv 3444 ⊆ wss 3911 Or wor 5545 ℩cio 6447 ‘cfv 6497 Isom wiso 6498 (class class class)co 7358 Fincfn 8886 ℝcr 11055 0cc0 11056 1c1 11057 + caddc 11059 < clt 11194 − cmin 11390 -cneg 11391 ℕ0cn0 12418 ...cfz 13430 ♯chash 14236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-hash 14237 |
This theorem is referenced by: fourierdlem50 44483 fourierdlem51 44484 fourierdlem52 44485 fourierdlem54 44487 fourierdlem76 44509 fourierdlem102 44535 fourierdlem103 44536 fourierdlem104 44537 fourierdlem114 44547 |
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