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Mirrors > Home > ILE Home > Th. List > iseqf1olemqf1o | GIF version |
Description: Lemma for seq3f1o 10507. 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 10494 | . . 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 10495 | . . . . 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 2559 | . . 3 ⊢ (𝜑 → ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄‘𝑣) = (𝑄‘𝑤) → 𝑣 = 𝑤)) |
13 | dff13 5772 | . . 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 10027 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | |
16 | 1, 15 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
17 | elfzel2 10026 | . . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | |
18 | 1, 17 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
19 | 16, 18 | fzfigd 10434 | . . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) |
20 | enrefg 6767 | . . . 4 ⊢ ((𝑀...𝑁) ∈ Fin → (𝑀...𝑁) ≈ (𝑀...𝑁)) | |
21 | 19, 20 | syl 14 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ≈ (𝑀...𝑁)) |
22 | f1finf1o 6949 | . . 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 1353 ∈ wcel 2148 ∀wral 2455 ifcif 3536 class class class wbr 4005 ↦ cmpt 4066 ◡ccnv 4627 ⟶wf 5214 –1-1→wf1 5215 –1-1-onto→wf1o 5217 ‘cfv 5218 (class class class)co 5878 ≈ cen 6741 Fincfn 6743 1c1 7815 − cmin 8131 ℤcz 9256 ...cfz 10011 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-frec 6395 df-1o 6420 df-er 6538 df-en 6744 df-fin 6746 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-inn 8923 df-n0 9180 df-z 9257 df-uz 9532 df-fz 10012 |
This theorem is referenced by: seq3f1olemqsumkj 10501 seq3f1olemqsumk 10502 seq3f1olemstep 10504 |
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