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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmtrto1cl | Structured version Visualization version GIF version |
Description: Useful lemma for the following theorems. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
Ref | Expression |
---|---|
psgnfzto1st.d | ⊢ 𝐷 = (1...𝑁) |
pmtrto1cl.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
Ref | Expression |
---|---|
pmtrto1cl | ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑇‘{𝐾, (𝐾 + 1)}) ∈ ran 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psgnfzto1st.d | . . . 4 ⊢ 𝐷 = (1...𝑁) | |
2 | fzfi 13973 | . . . 4 ⊢ (1...𝑁) ∈ Fin | |
3 | 1, 2 | eqeltri 2821 | . . 3 ⊢ 𝐷 ∈ Fin |
4 | 3 | a1i 11 | . 2 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝐷 ∈ Fin) |
5 | simpl 481 | . . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝐾 ∈ ℕ) | |
6 | simpr 483 | . . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝐾 + 1) ∈ 𝐷) | |
7 | 6, 1 | eleqtrdi 2835 | . . . . . . . 8 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝐾 + 1) ∈ (1...𝑁)) |
8 | elfz1b 13605 | . . . . . . . 8 ⊢ ((𝐾 + 1) ∈ (1...𝑁) ↔ ((𝐾 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝐾 + 1) ≤ 𝑁)) | |
9 | 7, 8 | sylib 217 | . . . . . . 7 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → ((𝐾 + 1) ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ (𝐾 + 1) ≤ 𝑁)) |
10 | 9 | simp2d 1140 | . . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝑁 ∈ ℕ) |
11 | 5 | nnred 12260 | . . . . . . 7 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝐾 ∈ ℝ) |
12 | 1red 11247 | . . . . . . . 8 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 1 ∈ ℝ) | |
13 | 11, 12 | readdcld 11275 | . . . . . . 7 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝐾 + 1) ∈ ℝ) |
14 | 10 | nnred 12260 | . . . . . . 7 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝑁 ∈ ℝ) |
15 | 11 | lep1d 12178 | . . . . . . 7 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝐾 ≤ (𝐾 + 1)) |
16 | 9 | simp3d 1141 | . . . . . . 7 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝐾 + 1) ≤ 𝑁) |
17 | 11, 13, 14, 15, 16 | letrd 11403 | . . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝐾 ≤ 𝑁) |
18 | 5, 10, 17 | 3jca 1125 | . . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁)) |
19 | elfz1b 13605 | . . . . 5 ⊢ (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐾 ≤ 𝑁)) | |
20 | 18, 19 | sylibr 233 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝐾 ∈ (1...𝑁)) |
21 | 20, 1 | eleqtrrdi 2836 | . . 3 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝐾 ∈ 𝐷) |
22 | prssi 4826 | . . 3 ⊢ ((𝐾 ∈ 𝐷 ∧ (𝐾 + 1) ∈ 𝐷) → {𝐾, (𝐾 + 1)} ⊆ 𝐷) | |
23 | 21, 6, 22 | syl2anc 582 | . 2 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → {𝐾, (𝐾 + 1)} ⊆ 𝐷) |
24 | 11 | ltp1d 12177 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝐾 < (𝐾 + 1)) |
25 | 11, 24 | ltned 11382 | . . 3 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → 𝐾 ≠ (𝐾 + 1)) |
26 | enpr2 10027 | . . 3 ⊢ ((𝐾 ∈ 𝐷 ∧ (𝐾 + 1) ∈ 𝐷 ∧ 𝐾 ≠ (𝐾 + 1)) → {𝐾, (𝐾 + 1)} ≈ 2o) | |
27 | 21, 6, 25, 26 | syl3anc 1368 | . 2 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → {𝐾, (𝐾 + 1)} ≈ 2o) |
28 | pmtrto1cl.t | . . 3 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
29 | eqid 2725 | . . 3 ⊢ ran 𝑇 = ran 𝑇 | |
30 | 28, 29 | pmtrrn 19424 | . 2 ⊢ ((𝐷 ∈ Fin ∧ {𝐾, (𝐾 + 1)} ⊆ 𝐷 ∧ {𝐾, (𝐾 + 1)} ≈ 2o) → (𝑇‘{𝐾, (𝐾 + 1)}) ∈ ran 𝑇) |
31 | 4, 23, 27, 30 | syl3anc 1368 | 1 ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑇‘{𝐾, (𝐾 + 1)}) ∈ ran 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ⊆ wss 3944 {cpr 4632 class class class wbr 5149 ran crn 5679 ‘cfv 6549 (class class class)co 7419 2oc2o 8481 ≈ cen 8961 Fincfn 8964 1c1 11141 + caddc 11143 ≤ cle 11281 ℕcn 12245 ...cfz 13519 pmTrspcpmtr 19408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-pmtr 19409 |
This theorem is referenced by: psgnfzto1stlem 32913 fzto1st 32916 psgnfzto1st 32918 |
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