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Theorem iseqf1olemqcl 10861

Description: Lemma for seq3f1o 10879. (Contributed by Jim Kingdon, 27-Aug-2022.)

**Hypotheses**
Ref	Expression
iseqf1olemqcl.k	⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
iseqf1olemqcl.j	⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1olemqcl.a	⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁))

**Assertion**
Ref	Expression
iseqf1olemqcl	⊢ (𝜑 → if(𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴)) ∈ (𝑀...𝑁))

**Proof of Theorem iseqf1olemqcl**
Step	Hyp	Ref	Expression
1		iseqf1olemqcl.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
2	1	ad2antrr 488	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ 𝐴 = 𝐾) → 𝐾 ∈ (𝑀...𝑁))
3		iseqf1olemqcl.j	. . . . . 6 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
4		f1of 5614	. . . . . 6 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
5	3, 4	syl 14	. . . . 5 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
6	5	ad2antrr 488	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
7	1	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ (𝑀...𝑁))
8		elfzel1 10358	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
9	7, 8	syl 14	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ∈ ℤ)
10		elfzel2 10357	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
11	7, 10	syl 14	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑁 ∈ ℤ)
12		iseqf1olemqcl.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁))
13		elfzelz 10359	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑀...𝑁) → 𝐴 ∈ ℤ)
14	12, 13	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
15	14	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℤ)
16		peano2zm 9615	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 − 1) ∈ ℤ)
17	15, 16	syl 14	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ ℤ)
18	9, 11, 17	3jca 1204	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ))
19	9	zred 9700	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ∈ ℝ)
20		elfzelz 10359	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
21	7, 20	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℤ)
22	21	zred 9700	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℝ)
23	17	zred 9700	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ ℝ)
24		elfzle1 10361	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾)
25	7, 24	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ≤ 𝐾)
26		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐴 = 𝐾)
27		eqcom 2234	. . . . . . . . . 10 ⊢ (𝐴 = 𝐾 ↔ 𝐾 = 𝐴)
28	26, 27	sylnib 683	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐾 = 𝐴)
29		elfzle1 10361	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)) → 𝐾 ≤ 𝐴)
30	29	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ≤ 𝐴)
31		zleloe 9624	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾 ≤ 𝐴 ↔ (𝐾 < 𝐴 ∨ 𝐾 = 𝐴)))
32	21, 15, 31	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 ≤ 𝐴 ↔ (𝐾 < 𝐴 ∨ 𝐾 = 𝐴)))
33	30, 32	mpbid 147	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 < 𝐴 ∨ 𝐾 = 𝐴))
34	28, 33	ecased 1386	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 < 𝐴)
35		zltlem1 9635	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾 < 𝐴 ↔ 𝐾 ≤ (𝐴 − 1)))
36	21, 15, 35	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 < 𝐴 ↔ 𝐾 ≤ (𝐴 − 1)))
37	34, 36	mpbid 147	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ≤ (𝐴 − 1))
38	19, 22, 23, 25, 37	letrd 8397	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ≤ (𝐴 − 1))
39	15	zred 9700	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℝ)
40	11	zred 9700	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑁 ∈ ℝ)
41	39	lem1d 9207	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ 𝐴)
42	12	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ (𝑀...𝑁))
43		elfzle2 10362	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑀...𝑁) → 𝐴 ≤ 𝑁)
44	42, 43	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ≤ 𝑁)
45	23, 39, 40, 41, 44	letrd 8397	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ 𝑁)
46	38, 45	jca 306	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝑀 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ 𝑁))
47		elfz2 10349	. . . . 5 ⊢ ((𝐴 − 1) ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ) ∧ (𝑀 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ 𝑁)))
48	18, 46, 47	sylanbrc 417	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ (𝑀...𝑁))
49	6, 48	ffvelcdmd 5813	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐽‘(𝐴 − 1)) ∈ (𝑀...𝑁))
50	1, 20	syl 14	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
51		zdceq 9653	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ) → DECID 𝐴 = 𝐾)
52	14, 50, 51	syl2anc 411	. . . 4 ⊢ (𝜑 → DECID 𝐴 = 𝐾)
53	52	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) → DECID 𝐴 = 𝐾)
54	2, 49, 53	ifcldadc 3652	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) → if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))) ∈ (𝑀...𝑁))
55	5, 12	ffvelcdmd 5813	. . 3 ⊢ (𝜑 → (𝐽‘𝐴) ∈ (𝑀...𝑁))
56	55	adantr 276	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) → (𝐽‘𝐴) ∈ (𝑀...𝑁))
57		f1ocnv 5627	. . . . . 6 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ^◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
58		f1of 5614	. . . . . 6 ⊢ (^◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ^◡𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
59	3, 57, 58	3syl 17	. . . . 5 ⊢ (𝜑 → ^◡𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
60	59, 1	ffvelcdmd 5813	. . . 4 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (𝑀...𝑁))
61		elfzelz 10359	. . . 4 ⊢ ((^◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (^◡𝐽‘𝐾) ∈ ℤ)
62	60, 61	syl 14	. . 3 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℤ)
63		fzdcel 10374	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ) → DECID 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)))
64	14, 50, 62, 63	syl3anc 1274	. 2 ⊢ (𝜑 → DECID 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)))
65	54, 56, 64	ifcldadc 3652	1 ⊢ (𝜑 → if(𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴)) ∈ (𝑀...𝑁))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 ∧ w3a 1005 = wceq 1398 ∈ wcel 2203 ifcif 3620 class class class wbr 4109 ^◡ccnv 4748 ⟶wf 5348 –1-1-onto→wf1o 5351 ‘cfv 5352 (class class class)co 6050 1c1 8128 < clt 8308 ≤ cle 8309 − cmin 8444 ℤcz 9577 ...cfz 10342

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-addcom 8227 ax-addass 8229 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-0id 8235 ax-rnegex 8236 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-ltadd 8243

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-inn 9238 df-n0 9497 df-z 9578 df-uz 9854 df-fz 10343

This theorem is referenced by: iseqf1olemqval 10862 iseqf1olemqf 10866

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