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Theorem iseqf1olemqcl 10681

Description: Lemma for seq3f1o 10699. (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 5544	. . . . . 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 10181	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
9	7, 8	syl 14	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ∈ ℤ)
10		elfzel2 10180	. . . . . . 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 10182	. . . . . . . . 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 9445	. . . . . . 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 1180	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ))
19	9	zred 9530	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ∈ ℝ)
20		elfzelz 10182	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
21	7, 20	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℤ)
22	21	zred 9530	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℝ)
23	17	zred 9530	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ ℝ)
24		elfzle1 10184	. . . . . . . 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 2209	. . . . . . . . . 10 ⊢ (𝐴 = 𝐾 ↔ 𝐾 = 𝐴)
28	26, 27	sylnib 678	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐾 = 𝐴)
29		elfzle1 10184	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)) → 𝐾 ≤ 𝐴)
30	29	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ≤ 𝐴)
31		zleloe 9454	. . . . . . . . . . 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 1362	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 < 𝐴)
35		zltlem1 9465	. . . . . . . . 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 8231	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ≤ (𝐴 − 1))
39	15	zred 9530	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℝ)
40	11	zred 9530	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑁 ∈ ℝ)
41	39	lem1d 9041	. . . . . . 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 10185	. . . . . . . 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 8231	. . . . . 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 10172	. . . . 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 5739	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐽‘(𝐴 − 1)) ∈ (𝑀...𝑁))
50	1, 20	syl 14	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
51		zdceq 9483	. . . . 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 3609	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) → if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))) ∈ (𝑀...𝑁))
55	5, 12	ffvelcdmd 5739	. . 3 ⊢ (𝜑 → (𝐽‘𝐴) ∈ (𝑀...𝑁))
56	55	adantr 276	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) → (𝐽‘𝐴) ∈ (𝑀...𝑁))
57		f1ocnv 5557	. . . . . 6 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ^◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
58		f1of 5544	. . . . . 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 5739	. . . 4 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (𝑀...𝑁))
61		elfzelz 10182	. . . 4 ⊢ ((^◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (^◡𝐽‘𝐾) ∈ ℤ)
62	60, 61	syl 14	. . 3 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℤ)
63		fzdcel 10197	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ) → DECID 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)))
64	14, 50, 62, 63	syl3anc 1250	. 2 ⊢ (𝜑 → DECID 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)))
65	54, 56, 64	ifcldadc 3609	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 710 DECID wdc 836 ∧ w3a 981 = wceq 1373 ∈ wcel 2178 ifcif 3579 class class class wbr 4059 ^◡ccnv 4692 ⟶wf 5286 –1-1-onto→wf1o 5289 ‘cfv 5290 (class class class)co 5967 1c1 7961 < clt 8142 ≤ cle 8143 − cmin 8278 ℤcz 9407 ...cfz 10165

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-addcom 8060 ax-addass 8062 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-0id 8068 ax-rnegex 8069 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-ltadd 8076

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-br 4060 df-opab 4122 df-mpt 4123 df-id 4358 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-inn 9072 df-n0 9331 df-z 9408 df-uz 9684 df-fz 10166

This theorem is referenced by: iseqf1olemqval 10682 iseqf1olemqf 10686

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