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Mirrors > Home > ILE Home > Th. List > iseqf1olemklt | GIF version |
Description: Lemma for seq3f1o 10245. (Contributed by Jim Kingdon, 21-Aug-2022.) |
Ref | Expression |
---|---|
iseqf1olemklt.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
iseqf1olemklt.k | ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) |
iseqf1olemklt.j | ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |
iseqf1olemklt.const | ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) |
iseqf1olemklt.kj | ⊢ (𝜑 → 𝐾 ≠ (◡𝐽‘𝐾)) |
Ref | Expression |
---|---|
iseqf1olemklt | ⊢ (𝜑 → 𝐾 < (◡𝐽‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqf1olemklt.kj | . . 3 ⊢ (𝜑 → 𝐾 ≠ (◡𝐽‘𝐾)) | |
2 | 1 | neneqd 2306 | . 2 ⊢ (𝜑 → ¬ 𝐾 = (◡𝐽‘𝐾)) |
3 | iseqf1olemklt.j | . . . . . 6 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) | |
4 | 3 | adantr 274 | . . . . 5 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |
5 | iseqf1olemklt.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) | |
6 | 5 | adantr 274 | . . . . 5 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → 𝐾 ∈ (𝑀...𝑁)) |
7 | f1ocnvfv2 5647 | . . . . 5 ⊢ ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽‘(◡𝐽‘𝐾)) = 𝐾) | |
8 | 4, 6, 7 | syl2anc 408 | . . . 4 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (𝐽‘(◡𝐽‘𝐾)) = 𝐾) |
9 | fveq2 5389 | . . . . . 6 ⊢ (𝑥 = (◡𝐽‘𝐾) → (𝐽‘𝑥) = (𝐽‘(◡𝐽‘𝐾))) | |
10 | id 19 | . . . . . 6 ⊢ (𝑥 = (◡𝐽‘𝐾) → 𝑥 = (◡𝐽‘𝐾)) | |
11 | 9, 10 | eqeq12d 2132 | . . . . 5 ⊢ (𝑥 = (◡𝐽‘𝐾) → ((𝐽‘𝑥) = 𝑥 ↔ (𝐽‘(◡𝐽‘𝐾)) = (◡𝐽‘𝐾))) |
12 | iseqf1olemklt.const | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) | |
13 | 12 | adantr 274 | . . . . 5 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) |
14 | f1ocnv 5348 | . . . . . . . . . . 11 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) | |
15 | 3, 14 | syl 14 | . . . . . . . . . 10 ⊢ (𝜑 → ◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |
16 | f1of 5335 | . . . . . . . . . 10 ⊢ (◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ◡𝐽:(𝑀...𝑁)⟶(𝑀...𝑁)) | |
17 | 15, 16 | syl 14 | . . . . . . . . 9 ⊢ (𝜑 → ◡𝐽:(𝑀...𝑁)⟶(𝑀...𝑁)) |
18 | 17, 5 | ffvelrnd 5524 | . . . . . . . 8 ⊢ (𝜑 → (◡𝐽‘𝐾) ∈ (𝑀...𝑁)) |
19 | elfzuz 9770 | . . . . . . . 8 ⊢ ((◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (◡𝐽‘𝐾) ∈ (ℤ≥‘𝑀)) | |
20 | 18, 19 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (◡𝐽‘𝐾) ∈ (ℤ≥‘𝑀)) |
21 | 20 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (◡𝐽‘𝐾) ∈ (ℤ≥‘𝑀)) |
22 | elfzelz 9774 | . . . . . . . 8 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ) | |
23 | 5, 22 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
24 | 23 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → 𝐾 ∈ ℤ) |
25 | simpr 109 | . . . . . 6 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (◡𝐽‘𝐾) < 𝐾) | |
26 | elfzo2 9895 | . . . . . 6 ⊢ ((◡𝐽‘𝐾) ∈ (𝑀..^𝐾) ↔ ((◡𝐽‘𝐾) ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ ∧ (◡𝐽‘𝐾) < 𝐾)) | |
27 | 21, 24, 25, 26 | syl3anbrc 1150 | . . . . 5 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (◡𝐽‘𝐾) ∈ (𝑀..^𝐾)) |
28 | 11, 13, 27 | rspcdva 2768 | . . . 4 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (𝐽‘(◡𝐽‘𝐾)) = (◡𝐽‘𝐾)) |
29 | 8, 28 | eqtr3d 2152 | . . 3 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → 𝐾 = (◡𝐽‘𝐾)) |
30 | 2, 29 | mtand 639 | . 2 ⊢ (𝜑 → ¬ (◡𝐽‘𝐾) < 𝐾) |
31 | elfzelz 9774 | . . . 4 ⊢ ((◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (◡𝐽‘𝐾) ∈ ℤ) | |
32 | 18, 31 | syl 14 | . . 3 ⊢ (𝜑 → (◡𝐽‘𝐾) ∈ ℤ) |
33 | ztri3or 9065 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ (◡𝐽‘𝐾) ∈ ℤ) → (𝐾 < (◡𝐽‘𝐾) ∨ 𝐾 = (◡𝐽‘𝐾) ∨ (◡𝐽‘𝐾) < 𝐾)) | |
34 | 23, 32, 33 | syl2anc 408 | . 2 ⊢ (𝜑 → (𝐾 < (◡𝐽‘𝐾) ∨ 𝐾 = (◡𝐽‘𝐾) ∨ (◡𝐽‘𝐾) < 𝐾)) |
35 | 2, 30, 34 | ecase23d 1313 | 1 ⊢ (𝜑 → 𝐾 < (◡𝐽‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ w3o 946 = wceq 1316 ∈ wcel 1465 ≠ wne 2285 ∀wral 2393 class class class wbr 3899 ◡ccnv 4508 ⟶wf 5089 –1-1-onto→wf1o 5092 ‘cfv 5093 (class class class)co 5742 < clt 7768 ℤcz 9022 ℤ≥cuz 9294 ...cfz 9758 ..^cfzo 9887 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 df-fz 9759 df-fzo 9888 |
This theorem is referenced by: seq3f1olemqsumkj 10239 |
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