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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 11339 | . . . . . . 7 ⊢ < Or ℝ | |
5 | soss 5617 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴)) | |
6 | 3, 4, 5 | mpisyl 21 | . . . . . 6 ⊢ (𝜑 → < Or 𝐴) |
7 | 0zd 12623 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℤ) | |
8 | eqid 2735 | . . . . . 6 ⊢ ((♯‘𝐴) + (0 − 1)) = ((♯‘𝐴) + (0 − 1)) | |
9 | 2, 6, 7, 8 | fzisoeu 45251 | . . . . 5 ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((0...((♯‘𝐴) + (0 − 1))), 𝐴)) |
10 | hashcl 14392 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
11 | 2, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
12 | 11 | nn0cnd 12587 | . . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ) |
13 | 1cnd 11254 | . . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ) | |
14 | 12, 13 | negsubd 11624 | . . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝐴) + -1) = ((♯‘𝐴) − 1)) |
15 | df-neg 11493 | . . . . . . . . . . 11 ⊢ -1 = (0 − 1) | |
16 | 15 | eqcomi 2744 | . . . . . . . . . 10 ⊢ (0 − 1) = -1 |
17 | 16 | oveq2i 7442 | . . . . . . . . 9 ⊢ ((♯‘𝐴) + (0 − 1)) = ((♯‘𝐴) + -1) |
18 | fourierdlem36.n | . . . . . . . . 9 ⊢ 𝑁 = ((♯‘𝐴) − 1) | |
19 | 14, 17, 18 | 3eqtr4g 2800 | . . . . . . . 8 ⊢ (𝜑 → ((♯‘𝐴) + (0 − 1)) = 𝑁) |
20 | 19 | oveq2d 7447 | . . . . . . 7 ⊢ (𝜑 → (0...((♯‘𝐴) + (0 − 1))) = (0...𝑁)) |
21 | isoeq4 7340 | . . . . . . 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 2584 | . . . . 5 ⊢ (𝜑 → (∃!𝑓 𝑓 Isom < , < ((0...((♯‘𝐴) + (0 − 1))), 𝐴) ↔ ∃!𝑓 𝑓 Isom < , < ((0...𝑁), 𝐴))) |
24 | 9, 23 | mpbid 232 | . . . 4 ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((0...𝑁), 𝐴)) |
25 | iotacl 6549 | . . . 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 2843 | . 2 ⊢ (𝜑 → 𝐹 ∈ {𝑓 ∣ 𝑓 Isom < , < ((0...𝑁), 𝐴)}) |
28 | iotaex 6536 | . . . 4 ⊢ (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴)) ∈ V | |
29 | 1, 28 | eqeltri 2835 | . . 3 ⊢ 𝐹 ∈ V |
30 | isoeq1 7337 | . . 3 ⊢ (𝑓 = 𝐹 → (𝑓 Isom < , < ((0...𝑁), 𝐴) ↔ 𝐹 Isom < , < ((0...𝑁), 𝐴))) | |
31 | 29, 30 | elab 3681 | . 2 ⊢ (𝐹 ∈ {𝑓 ∣ 𝑓 Isom < , < ((0...𝑁), 𝐴)} ↔ 𝐹 Isom < , < ((0...𝑁), 𝐴)) |
32 | 27, 31 | sylib 218 | 1 ⊢ (𝜑 → 𝐹 Isom < , < ((0...𝑁), 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∃!weu 2566 {cab 2712 Vcvv 3478 ⊆ wss 3963 Or wor 5596 ℩cio 6514 ‘cfv 6563 Isom wiso 6564 (class class class)co 7431 Fincfn 8984 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 < clt 11293 − cmin 11490 -cneg 11491 ℕ0cn0 12524 ...cfz 13544 ♯chash 14366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 |
This theorem is referenced by: fourierdlem50 46112 fourierdlem51 46113 fourierdlem52 46114 fourierdlem54 46116 fourierdlem76 46138 fourierdlem102 46164 fourierdlem103 46165 fourierdlem104 46166 fourierdlem114 46176 |
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