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 10707 | . . . . . . 7 ⊢ < Or ℝ | |
5 | soss 5479 | . . . . . . 7 ⊢ (𝐴 ⊆ ℝ → ( < Or ℝ → < Or 𝐴)) | |
6 | 3, 4, 5 | mpisyl 21 | . . . . . 6 ⊢ (𝜑 → < Or 𝐴) |
7 | 0zd 11980 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℤ) | |
8 | eqid 2821 | . . . . . 6 ⊢ ((♯‘𝐴) + (0 − 1)) = ((♯‘𝐴) + (0 − 1)) | |
9 | 2, 6, 7, 8 | fzisoeu 41657 | . . . . 5 ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((0...((♯‘𝐴) + (0 − 1))), 𝐴)) |
10 | hashcl 13707 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
11 | 2, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
12 | 11 | nn0cnd 11944 | . . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ) |
13 | 1cnd 10622 | . . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ) | |
14 | 12, 13 | negsubd 10989 | . . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝐴) + -1) = ((♯‘𝐴) − 1)) |
15 | df-neg 10859 | . . . . . . . . . . 11 ⊢ -1 = (0 − 1) | |
16 | 15 | eqcomi 2830 | . . . . . . . . . 10 ⊢ (0 − 1) = -1 |
17 | 16 | oveq2i 7153 | . . . . . . . . 9 ⊢ ((♯‘𝐴) + (0 − 1)) = ((♯‘𝐴) + -1) |
18 | fourierdlem36.n | . . . . . . . . 9 ⊢ 𝑁 = ((♯‘𝐴) − 1) | |
19 | 14, 17, 18 | 3eqtr4g 2881 | . . . . . . . 8 ⊢ (𝜑 → ((♯‘𝐴) + (0 − 1)) = 𝑁) |
20 | 19 | oveq2d 7158 | . . . . . . 7 ⊢ (𝜑 → (0...((♯‘𝐴) + (0 − 1))) = (0...𝑁)) |
21 | isoeq4 7059 | . . . . . . 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 2672 | . . . . 5 ⊢ (𝜑 → (∃!𝑓 𝑓 Isom < , < ((0...((♯‘𝐴) + (0 − 1))), 𝐴) ↔ ∃!𝑓 𝑓 Isom < , < ((0...𝑁), 𝐴))) |
24 | 9, 23 | mpbid 234 | . . . 4 ⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((0...𝑁), 𝐴)) |
25 | iotacl 6327 | . . . 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 2917 | . 2 ⊢ (𝜑 → 𝐹 ∈ {𝑓 ∣ 𝑓 Isom < , < ((0...𝑁), 𝐴)}) |
28 | iotaex 6321 | . . . 4 ⊢ (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴)) ∈ V | |
29 | 1, 28 | eqeltri 2909 | . . 3 ⊢ 𝐹 ∈ V |
30 | isoeq1 7056 | . . 3 ⊢ (𝑓 = 𝐹 → (𝑓 Isom < , < ((0...𝑁), 𝐴) ↔ 𝐹 Isom < , < ((0...𝑁), 𝐴))) | |
31 | 29, 30 | elab 3658 | . 2 ⊢ (𝐹 ∈ {𝑓 ∣ 𝑓 Isom < , < ((0...𝑁), 𝐴)} ↔ 𝐹 Isom < , < ((0...𝑁), 𝐴)) |
32 | 27, 31 | sylib 220 | 1 ⊢ (𝜑 → 𝐹 Isom < , < ((0...𝑁), 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃!weu 2653 {cab 2799 Vcvv 3486 ⊆ wss 3924 Or wor 5459 ℩cio 6298 ‘cfv 6341 Isom wiso 6342 (class class class)co 7142 Fincfn 8495 ℝcr 10522 0cc0 10523 1c1 10524 + caddc 10526 < clt 10661 − cmin 10856 -cneg 10857 ℕ0cn0 11884 ...cfz 12882 ♯chash 13680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-oi 8960 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-n0 11885 df-z 11969 df-uz 12231 df-fz 12883 df-hash 13681 |
This theorem is referenced by: fourierdlem50 42531 fourierdlem51 42532 fourierdlem52 42533 fourierdlem54 42535 fourierdlem76 42557 fourierdlem102 42583 fourierdlem103 42584 fourierdlem104 42585 fourierdlem114 42595 |
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