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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 11316 | . . . . . . 7 ⊢ < Or ℝ | |
5 | soss 5604 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴)) | |
6 | 3, 4, 5 | mpisyl 21 | . . . . . 6 ⊢ (𝜑 → < Or 𝐴) |
7 | 0zd 12592 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℤ) | |
8 | eqid 2727 | . . . . . 6 ⊢ ((♯‘𝐴) + (0 − 1)) = ((♯‘𝐴) + (0 − 1)) | |
9 | 2, 6, 7, 8 | fzisoeu 44605 | . . . . 5 ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((0...((♯‘𝐴) + (0 − 1))), 𝐴)) |
10 | hashcl 14339 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
11 | 2, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
12 | 11 | nn0cnd 12556 | . . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ) |
13 | 1cnd 11231 | . . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ) | |
14 | 12, 13 | negsubd 11599 | . . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝐴) + -1) = ((♯‘𝐴) − 1)) |
15 | df-neg 11469 | . . . . . . . . . . 11 ⊢ -1 = (0 − 1) | |
16 | 15 | eqcomi 2736 | . . . . . . . . . 10 ⊢ (0 − 1) = -1 |
17 | 16 | oveq2i 7425 | . . . . . . . . 9 ⊢ ((♯‘𝐴) + (0 − 1)) = ((♯‘𝐴) + -1) |
18 | fourierdlem36.n | . . . . . . . . 9 ⊢ 𝑁 = ((♯‘𝐴) − 1) | |
19 | 14, 17, 18 | 3eqtr4g 2792 | . . . . . . . 8 ⊢ (𝜑 → ((♯‘𝐴) + (0 − 1)) = 𝑁) |
20 | 19 | oveq2d 7430 | . . . . . . 7 ⊢ (𝜑 → (0...((♯‘𝐴) + (0 − 1))) = (0...𝑁)) |
21 | isoeq4 7322 | . . . . . . 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 2575 | . . . . 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 6528 | . . . 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 2832 | . 2 ⊢ (𝜑 → 𝐹 ∈ {𝑓 ∣ 𝑓 Isom < , < ((0...𝑁), 𝐴)}) |
28 | iotaex 6515 | . . . 4 ⊢ (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴)) ∈ V | |
29 | 1, 28 | eqeltri 2824 | . . 3 ⊢ 𝐹 ∈ V |
30 | isoeq1 7319 | . . 3 ⊢ (𝑓 = 𝐹 → (𝑓 Isom < , < ((0...𝑁), 𝐴) ↔ 𝐹 Isom < , < ((0...𝑁), 𝐴))) | |
31 | 29, 30 | elab 3665 | . 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 1534 ∈ wcel 2099 ∃!weu 2557 {cab 2704 Vcvv 3469 ⊆ wss 3944 Or wor 5583 ℩cio 6492 ‘cfv 6542 Isom wiso 6543 (class class class)co 7414 Fincfn 8955 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 < clt 11270 − cmin 11466 -cneg 11467 ℕ0cn0 12494 ...cfz 13508 ♯chash 14313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-hash 14314 |
This theorem is referenced by: fourierdlem50 45467 fourierdlem51 45468 fourierdlem52 45469 fourierdlem54 45471 fourierdlem76 45493 fourierdlem102 45519 fourierdlem103 45520 fourierdlem104 45521 fourierdlem114 45531 |
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