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Mirrors > Home > ILE Home > Th. List > iseqf1olemklt | GIF version |
Description: Lemma for seq3f1o 10501. (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 2368 | . 2 ⊢ (𝜑 → ¬ 𝐾 = (◡𝐽‘𝐾)) |
3 | iseqf1olemklt.j | . . . . . 6 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) | |
4 | 3 | adantr 276 | . . . . 5 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |
5 | iseqf1olemklt.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) | |
6 | 5 | adantr 276 | . . . . 5 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → 𝐾 ∈ (𝑀...𝑁)) |
7 | f1ocnvfv2 5778 | . . . . 5 ⊢ ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽‘(◡𝐽‘𝐾)) = 𝐾) | |
8 | 4, 6, 7 | syl2anc 411 | . . . 4 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (𝐽‘(◡𝐽‘𝐾)) = 𝐾) |
9 | fveq2 5515 | . . . . . 6 ⊢ (𝑥 = (◡𝐽‘𝐾) → (𝐽‘𝑥) = (𝐽‘(◡𝐽‘𝐾))) | |
10 | id 19 | . . . . . 6 ⊢ (𝑥 = (◡𝐽‘𝐾) → 𝑥 = (◡𝐽‘𝐾)) | |
11 | 9, 10 | eqeq12d 2192 | . . . . 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 276 | . . . . 5 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) |
14 | f1ocnv 5474 | . . . . . . . . . . 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 5461 | . . . . . . . . . 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 | ffvelcdmd 5652 | . . . . . . . 8 ⊢ (𝜑 → (◡𝐽‘𝐾) ∈ (𝑀...𝑁)) |
19 | elfzuz 10018 | . . . . . . . 8 ⊢ ((◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (◡𝐽‘𝐾) ∈ (ℤ≥‘𝑀)) | |
20 | 18, 19 | syl 14 | . . . . . . 7 ⊢ (𝜑 → (◡𝐽‘𝐾) ∈ (ℤ≥‘𝑀)) |
21 | 20 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (◡𝐽‘𝐾) ∈ (ℤ≥‘𝑀)) |
22 | elfzelz 10022 | . . . . . . . 8 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ) | |
23 | 5, 22 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
24 | 23 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → 𝐾 ∈ ℤ) |
25 | simpr 110 | . . . . . 6 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (◡𝐽‘𝐾) < 𝐾) | |
26 | elfzo2 10147 | . . . . . 6 ⊢ ((◡𝐽‘𝐾) ∈ (𝑀..^𝐾) ↔ ((◡𝐽‘𝐾) ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ ∧ (◡𝐽‘𝐾) < 𝐾)) | |
27 | 21, 24, 25, 26 | syl3anbrc 1181 | . . . . 5 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (◡𝐽‘𝐾) ∈ (𝑀..^𝐾)) |
28 | 11, 13, 27 | rspcdva 2846 | . . . 4 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (𝐽‘(◡𝐽‘𝐾)) = (◡𝐽‘𝐾)) |
29 | 8, 28 | eqtr3d 2212 | . . 3 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → 𝐾 = (◡𝐽‘𝐾)) |
30 | 2, 29 | mtand 665 | . 2 ⊢ (𝜑 → ¬ (◡𝐽‘𝐾) < 𝐾) |
31 | elfzelz 10022 | . . . 4 ⊢ ((◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (◡𝐽‘𝐾) ∈ ℤ) | |
32 | 18, 31 | syl 14 | . . 3 ⊢ (𝜑 → (◡𝐽‘𝐾) ∈ ℤ) |
33 | ztri3or 9294 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ (◡𝐽‘𝐾) ∈ ℤ) → (𝐾 < (◡𝐽‘𝐾) ∨ 𝐾 = (◡𝐽‘𝐾) ∨ (◡𝐽‘𝐾) < 𝐾)) | |
34 | 23, 32, 33 | syl2anc 411 | . 2 ⊢ (𝜑 → (𝐾 < (◡𝐽‘𝐾) ∨ 𝐾 = (◡𝐽‘𝐾) ∨ (◡𝐽‘𝐾) < 𝐾)) |
35 | 2, 30, 34 | ecase23d 1350 | 1 ⊢ (𝜑 → 𝐾 < (◡𝐽‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 class class class wbr 4003 ◡ccnv 4625 ⟶wf 5212 –1-1-onto→wf1o 5215 ‘cfv 5216 (class class class)co 5874 < clt 7990 ℤcz 9251 ℤ≥cuz 9526 ...cfz 10006 ..^cfzo 10139 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-n0 9175 df-z 9252 df-uz 9527 df-fz 10007 df-fzo 10140 |
This theorem is referenced by: seq3f1olemqsumkj 10495 |
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