Theorem iseqf1olemqf1o 10513

Description: Lemma for seq3f1o 10524. 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.)

Hypotheses

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)))

Assertion

Ref	Expression
iseqf1olemqf1o	⊢ (𝜑 → 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))

Distinct variable groups:   𝑢,𝐽   𝑢,𝐾   𝑢,𝑀   𝑢,𝑁   𝜑,𝑢

Allowed substitution hint:   𝑄(𝑢)

Proof of Theorem iseqf1olemqf1o

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

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 10511	. . 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 10512	. . . . 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 2572	. . 3 ⊢ (𝜑 → ∀𝑣 ∈ (𝑀...𝑁)∀𝑤 ∈ (𝑀...𝑁)((𝑄‘𝑣) = (𝑄‘𝑤) → 𝑣 = 𝑤))
13		dff13 5786	. . 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 10044	. . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
16	1, 15	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
17		elfzel2 10043	. . . . . 6 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
18	1, 17	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
19	16, 18	fzfigd 10451	. . . 4 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
20		enrefg 6783	. . . 4 ⊢ ((𝑀...𝑁) ∈ Fin → (𝑀...𝑁) ≈ (𝑀...𝑁))
21	19, 20	syl 14	. . 3 ⊢ (𝜑 → (𝑀...𝑁) ≈ (𝑀...𝑁))
22		f1finf1o 6966	. . 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))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2160  ∀wral 2468  ifcif 3549   class class class wbr 4018   ↦ cmpt 4079  ^◡ccnv 4640  ⟶wf 5228  –1-1→wf1 5229  –1-1-onto→wf1o 5231  ‘cfv 5232  (class class class)co 5892   ≈ cen 6757  Fincfn 6759  1c1 7832   − cmin 8148  ℤcz 9273  ...cfz 10028

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-addcom 7931  ax-addass 7933  ax-distr 7935  ax-i2m1 7936  ax-0lt1 7937  ax-0id 7939  ax-rnegex 7940  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-ltwlin 7944  ax-pre-lttrn 7945  ax-pre-apti 7946  ax-pre-ltadd 7947

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-frec 6411  df-1o 6436  df-er 6554  df-en 6760  df-fin 6762  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018  df-sub 8150  df-neg 8151  df-inn 8940  df-n0 9197  df-z 9274  df-uz 9549  df-fz 10029

This theorem is referenced by:  seq3f1olemqsumkj  10518  seq3f1olemqsumk  10519  seq3f1olemstep  10521