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Mirrors > Home > ILE Home > Th. List > iseqf1olemqf1o | GIF version |
Description: Lemma for seq3f1o 10517. 𝑄 is a permutation of (𝑀...𝑁). 𝑄 is formed from the constant portion of 𝐽, followed by the single element 𝐾 (at position 𝐾), followed by the rest of J (with the 𝐾 deleted and the elements before 𝐾 moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.) |
Ref | Expression |
---|---|
iseqf1olemqf.k | ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) |
iseqf1olemqf.j | ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |
iseqf1olemqf.q | ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) |
Ref | Expression |
---|---|
iseqf1olemqf1o | ⊢ (𝜑 → 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqf1olemqf.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) | |
2 | iseqf1olemqf.j | . . . 4 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) | |
3 | iseqf1olemqf.q | . . . 4 ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) | |
4 | 1, 2, 3 | iseqf1olemqf 10504 | . . 3 ⊢ (𝜑 → 𝑄:(𝑀...𝑁)⟶(𝑀...𝑁)) |
5 | 1 | ad2antrr 488 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝐾 ∈ (𝑀...𝑁)) |
6 | 2 | ad2antrr 488 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |
7 | simplrl 535 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝑣 ∈ (𝑀...𝑁)) | |
8 | simplrr 536 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝑤 ∈ (𝑀...𝑁)) | |
9 | simpr 110 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → (𝑄‘𝑣) = (𝑄‘𝑤)) | |
10 | 5, 6, 3, 7, 8, 9 | iseqf1olemmo 10505 | . . . . 5 ⊢ (((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) ∧ (𝑄‘𝑣) = (𝑄‘𝑤)) → 𝑣 = 𝑤) |
11 | 10 | ex 115 | . . . 4 ⊢ ((𝜑 ∧ (𝑣 ∈ (𝑀...𝑁) ∧ 𝑤 ∈ (𝑀...𝑁))) → ((𝑄‘𝑣) = (𝑄‘𝑤) → 𝑣 = 𝑤)) |
12 | 11 | ralrimivva 2569 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄‘𝑣) = (𝑄‘𝑤) → 𝑣 = 𝑤)) |
13 | dff13 5782 | . . 3 ⊢ (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ (𝑄:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄‘𝑣) = (𝑄‘𝑤) → 𝑣 = 𝑤))) | |
14 | 4, 12, 13 | sylanbrc 417 | . 2 ⊢ (𝜑 → 𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁)) |
15 | elfzel1 10037 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
16 | 1, 15 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
17 | elfzel2 10036 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | |
18 | 1, 17 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
19 | 16, 18 | fzfigd 10444 | . . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) |
20 | enrefg 6777 | . . . 4 ⊢ ((𝑀...𝑁) ∈ Fin → (𝑀...𝑁) ≈ (𝑀...𝑁)) | |
21 | 19, 20 | syl 14 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ≈ (𝑀...𝑁)) |
22 | f1finf1o 6959 | . . 3 ⊢ (((𝑀...𝑁) ≈ (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))) | |
23 | 21, 19, 22 | syl2anc 411 | . 2 ⊢ (𝜑 → (𝑄:(𝑀...𝑁)–1-1→(𝑀...𝑁) ↔ 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))) |
24 | 14, 23 | mpbid 147 | 1 ⊢ (𝜑 → 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 ∈ wcel 2158 ∀wral 2465 ifcif 3546 class class class wbr 4015 ↦ cmpt 4076 ◡ccnv 4637 ⟶wf 5224 –1-1→wf1 5225 –1-1-onto→wf1o 5227 ‘cfv 5228 (class class class)co 5888 ≈ cen 6751 Fincfn 6753 1c1 7825 − cmin 8141 ℤcz 9266 ...cfz 10021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-1o 6430 df-er 6548 df-en 6754 df-fin 6756 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-n0 9190 df-z 9267 df-uz 9542 df-fz 10022 |
This theorem is referenced by: seq3f1olemqsumkj 10511 seq3f1olemqsumk 10512 seq3f1olemstep 10514 |
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