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Mirrors > Home > MPE Home > Th. List > Mathboxes > reprfi2 | Structured version Visualization version GIF version |
Description: Corollary of reprinfz1 32185. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
Ref | Expression |
---|---|
reprinfz1.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
reprinfz1.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
reprinfz1.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ) |
Ref | Expression |
---|---|
reprfi2 | ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑁) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reprinfz1.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
2 | reprinfz1.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
3 | reprinfz1.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
4 | 1, 2, 3 | reprinfz1 32185 | . 2 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑁) = ((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑁)) |
5 | inss2 4130 | . . . . 5 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁) | |
6 | fz1ssnn 13042 | . . . . 5 ⊢ (1...𝑁) ⊆ ℕ | |
7 | 5, 6 | sstri 3896 | . . . 4 ⊢ (𝐴 ∩ (1...𝑁)) ⊆ ℕ |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 ∩ (1...𝑁)) ⊆ ℕ) |
9 | 1 | nn0zd 12179 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
10 | fzfi 13444 | . . . . 5 ⊢ (1...𝑁) ∈ Fin | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...𝑁) ∈ Fin) |
12 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (1...𝑁)) ⊆ (1...𝑁)) |
13 | 11, 12 | ssfid 8832 | . . 3 ⊢ (𝜑 → (𝐴 ∩ (1...𝑁)) ∈ Fin) |
14 | 8, 9, 2, 13 | reprfi 32179 | . 2 ⊢ (𝜑 → ((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑁) ∈ Fin) |
15 | 4, 14 | eqeltrd 2834 | 1 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑁) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∩ cin 3852 ⊆ wss 3853 ‘cfv 6350 (class class class)co 7183 Fincfn 8568 1c1 10629 ℕcn 11729 ℕ0cn0 11989 ...cfz 12994 reprcrepr 32171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-inf2 9190 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-pm 8453 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-sup 8992 df-oi 9060 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-n0 11990 df-z 12076 df-uz 12338 df-rp 12486 df-ico 12840 df-fz 12995 df-fzo 13138 df-seq 13474 df-exp 13535 df-hash 13796 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-clim 14948 df-sum 15149 df-repr 32172 |
This theorem is referenced by: hgt750lemb 32219 hgt750lema 32220 hgt750leme 32221 tgoldbachgtde 32223 tgoldbachgtda 32224 |
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