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Mirrors > Home > ILE Home > Th. List > iseqf1olemklt | GIF version |
Description: Lemma for seq3f1o 10277. (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 2329 | . 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 5679 | . . . . 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 5421 | . . . . . 6 ⊢ (𝑥 = (◡𝐽‘𝐾) → (𝐽‘𝑥) = (𝐽‘(◡𝐽‘𝐾))) | |
10 | id 19 | . . . . . 6 ⊢ (𝑥 = (◡𝐽‘𝐾) → 𝑥 = (◡𝐽‘𝐾)) | |
11 | 9, 10 | eqeq12d 2154 | . . . . 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 5380 | . . . . . . . . . . 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 5367 | . . . . . . . . . 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 5556 | . . . . . . . 8 ⊢ (𝜑 → (◡𝐽‘𝐾) ∈ (𝑀...𝑁)) |
19 | elfzuz 9802 | . . . . . . . 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 9806 | . . . . . . . 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 9927 | . . . . . 6 ⊢ ((◡𝐽‘𝐾) ∈ (𝑀..^𝐾) ↔ ((◡𝐽‘𝐾) ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ ∧ (◡𝐽‘𝐾) < 𝐾)) | |
27 | 21, 24, 25, 26 | syl3anbrc 1165 | . . . . 5 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (◡𝐽‘𝐾) ∈ (𝑀..^𝐾)) |
28 | 11, 13, 27 | rspcdva 2794 | . . . 4 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (𝐽‘(◡𝐽‘𝐾)) = (◡𝐽‘𝐾)) |
29 | 8, 28 | eqtr3d 2174 | . . 3 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → 𝐾 = (◡𝐽‘𝐾)) |
30 | 2, 29 | mtand 654 | . 2 ⊢ (𝜑 → ¬ (◡𝐽‘𝐾) < 𝐾) |
31 | elfzelz 9806 | . . . 4 ⊢ ((◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (◡𝐽‘𝐾) ∈ ℤ) | |
32 | 18, 31 | syl 14 | . . 3 ⊢ (𝜑 → (◡𝐽‘𝐾) ∈ ℤ) |
33 | ztri3or 9097 | . . 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 1328 | 1 ⊢ (𝜑 → 𝐾 < (◡𝐽‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ∀wral 2416 class class class wbr 3929 ◡ccnv 4538 ⟶wf 5119 –1-1-onto→wf1o 5122 ‘cfv 5123 (class class class)co 5774 < clt 7800 ℤcz 9054 ℤ≥cuz 9326 ...cfz 9790 ..^cfzo 9919 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-fzo 9920 |
This theorem is referenced by: seq3f1olemqsumkj 10271 |
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