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Theorem iseqf1olemqcl 10421

Description: Lemma for seq3f1o 10439. (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 480	. . 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 5432	. . . . . 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 480	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
7	1	ad2antrr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ (𝑀...𝑁))
8		elfzel1 9959	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
9	7, 8	syl 14	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ∈ ℤ)
10		elfzel2 9958	. . . . . . 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 9960	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑀...𝑁) → 𝐴 ∈ ℤ)
14	12, 13	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
15	14	ad2antrr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℤ)
16		peano2zm 9229	. . . . . . 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 1167	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ))
19	9	zred 9313	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ∈ ℝ)
20		elfzelz 9960	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
21	7, 20	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℤ)
22	21	zred 9313	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℝ)
23	17	zred 9313	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ ℝ)
24		elfzle1 9962	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾)
25	7, 24	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ≤ 𝐾)
26		simpr 109	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐴 = 𝐾)
27		eqcom 2167	. . . . . . . . . 10 ⊢ (𝐴 = 𝐾 ↔ 𝐾 = 𝐴)
28	26, 27	sylnib 666	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐾 = 𝐴)
29		elfzle1 9962	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)) → 𝐾 ≤ 𝐴)
30	29	ad2antlr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ≤ 𝐴)
31		zleloe 9238	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾 ≤ 𝐴 ↔ (𝐾 < 𝐴 ∨ 𝐾 = 𝐴)))
32	21, 15, 31	syl2anc 409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 ≤ 𝐴 ↔ (𝐾 < 𝐴 ∨ 𝐾 = 𝐴)))
33	30, 32	mpbid 146	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 < 𝐴 ∨ 𝐾 = 𝐴))
34	28, 33	ecased 1339	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 < 𝐴)
35		zltlem1 9248	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐾 < 𝐴 ↔ 𝐾 ≤ (𝐴 − 1)))
36	21, 15, 35	syl2anc 409	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐾 < 𝐴 ↔ 𝐾 ≤ (𝐴 − 1)))
37	34, 36	mpbid 146	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ≤ (𝐴 − 1))
38	19, 22, 23, 25, 37	letrd 8022	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ≤ (𝐴 − 1))
39	15	zred 9313	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℝ)
40	11	zred 9313	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑁 ∈ ℝ)
41	39	lem1d 8828	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ 𝐴)
42	12	ad2antrr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ (𝑀...𝑁))
43		elfzle2 9963	. . . . . . . 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 8022	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ≤ 𝑁)
46	38, 45	jca 304	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝑀 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ 𝑁))
47		elfz2 9951	. . . . 5 ⊢ ((𝐴 − 1) ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ) ∧ (𝑀 ≤ (𝐴 − 1) ∧ (𝐴 − 1) ≤ 𝑁)))
48	18, 46, 47	sylanbrc 414	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ (𝑀...𝑁))
49	6, 48	ffvelrnd 5621	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐽‘(𝐴 − 1)) ∈ (𝑀...𝑁))
50	1, 20	syl 14	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
51		zdceq 9266	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ) → DECID 𝐴 = 𝐾)
52	14, 50, 51	syl2anc 409	. . . 4 ⊢ (𝜑 → DECID 𝐴 = 𝐾)
53	52	adantr 274	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) → DECID 𝐴 = 𝐾)
54	2, 49, 53	ifcldadc 3549	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) → if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))) ∈ (𝑀...𝑁))
55	5, 12	ffvelrnd 5621	. . 3 ⊢ (𝜑 → (𝐽‘𝐴) ∈ (𝑀...𝑁))
56	55	adantr 274	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) → (𝐽‘𝐴) ∈ (𝑀...𝑁))
57		f1ocnv 5445	. . . . . 6 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ^◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
58		f1of 5432	. . . . . 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	ffvelrnd 5621	. . . 4 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (𝑀...𝑁))
61		elfzelz 9960	. . . 4 ⊢ ((^◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (^◡𝐽‘𝐾) ∈ ℤ)
62	60, 61	syl 14	. . 3 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℤ)
63		fzdcel 9975	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ) → DECID 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)))
64	14, 50, 62, 63	syl3anc 1228	. 2 ⊢ (𝜑 → DECID 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)))
65	54, 56, 64	ifcldadc 3549	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 103 ↔ wb 104 ∨ wo 698 DECID wdc 824 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 ifcif 3520 class class class wbr 3982 ^◡ccnv 4603 ⟶wf 5184 –1-1-onto→wf1o 5187 ‘cfv 5188 (class class class)co 5842 1c1 7754 < clt 7933 ≤ cle 7934 − cmin 8069 ℤcz 9191 ...cfz 9944

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-addass 7855 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-0id 7861 ax-rnegex 7862 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-ltadd 7869

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-inn 8858 df-n0 9115 df-z 9192 df-uz 9467 df-fz 9945

This theorem is referenced by: iseqf1olemqval 10422 iseqf1olemqf 10426

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