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Theorem iseqf1olemqcl 10644

Description: Lemma for seq3f1o 10662. (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 5522	. . . . . 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 10146	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
9	7, 8	syl 14	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ∈ ℤ)
10		elfzel2 10145	. . . . . . 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 10147	. . . . . . . . 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 9410	. . . . . . 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 9495	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ∈ ℝ)
20		elfzelz 10147	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
21	7, 20	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℤ)
22	21	zred 9495	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ∈ ℝ)
23	17	zred 9495	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐴 − 1) ∈ ℝ)
24		elfzle1 10149	. . . . . . . 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 2207	. . . . . . . . . 10 ⊢ (𝐴 = 𝐾 ↔ 𝐾 = 𝐴)
28	26, 27	sylnib 678	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → ¬ 𝐾 = 𝐴)
29		elfzle1 10149	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐾...(^◡𝐽‘𝐾)) → 𝐾 ≤ 𝐴)
30	29	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐾 ≤ 𝐴)
31		zleloe 9419	. . . . . . . . . . 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 9430	. . . . . . . . 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 8196	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑀 ≤ (𝐴 − 1))
39	15	zred 9495	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝐴 ∈ ℝ)
40	11	zred 9495	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → 𝑁 ∈ ℝ)
41	39	lem1d 9006	. . . . . . 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 10150	. . . . . . . 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 8196	. . . . . 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 10137	. . . . 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 5716	. . 3 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) ∧ ¬ 𝐴 = 𝐾) → (𝐽‘(𝐴 − 1)) ∈ (𝑀...𝑁))
50	1, 20	syl 14	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ)
51		zdceq 9448	. . . . 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 3600	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) → if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))) ∈ (𝑀...𝑁))
55	5, 12	ffvelcdmd 5716	. . 3 ⊢ (𝜑 → (𝐽‘𝐴) ∈ (𝑀...𝑁))
56	55	adantr 276	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ (𝐾...(^◡𝐽‘𝐾))) → (𝐽‘𝐴) ∈ (𝑀...𝑁))
57		f1ocnv 5535	. . . . . 6 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ^◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
58		f1of 5522	. . . . . 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 5716	. . . 4 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (𝑀...𝑁))
61		elfzelz 10147	. . . 4 ⊢ ((^◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (^◡𝐽‘𝐾) ∈ ℤ)
62	60, 61	syl 14	. . 3 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℤ)
63		fzdcel 10162	. . 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 3600	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 2176 ifcif 3571 class class class wbr 4044 ^◡ccnv 4674 ⟶wf 5267 –1-1-onto→wf1o 5270 ‘cfv 5271 (class class class)co 5944 1c1 7926 < clt 8107 ≤ cle 8108 − cmin 8243 ℤcz 9372 ...cfz 10130

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-addcom 8025 ax-addass 8027 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-0id 8033 ax-rnegex 8034 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-ltadd 8041

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 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-inn 9037 df-n0 9296 df-z 9373 df-uz 9649 df-fz 10131

This theorem is referenced by: iseqf1olemqval 10645 iseqf1olemqf 10649

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