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Mirrors > Home > ILE Home > Th. List > iseqf1olemklt | GIF version |
Description: Lemma for seq3f1o 10460. (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 2361 | . 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 5757 | . . . . 5 ⊢ ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽‘(◡𝐽‘𝐾)) = 𝐾) | |
8 | 4, 6, 7 | syl2anc 409 | . . . 4 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (𝐽‘(◡𝐽‘𝐾)) = 𝐾) |
9 | fveq2 5496 | . . . . . 6 ⊢ (𝑥 = (◡𝐽‘𝐾) → (𝐽‘𝑥) = (𝐽‘(◡𝐽‘𝐾))) | |
10 | id 19 | . . . . . 6 ⊢ (𝑥 = (◡𝐽‘𝐾) → 𝑥 = (◡𝐽‘𝐾)) | |
11 | 9, 10 | eqeq12d 2185 | . . . . 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 5455 | . . . . . . . . . . 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 5442 | . . . . . . . . . 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 5632 | . . . . . . . 8 ⊢ (𝜑 → (◡𝐽‘𝐾) ∈ (𝑀...𝑁)) |
19 | elfzuz 9977 | . . . . . . . 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 9981 | . . . . . . . 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 10106 | . . . . . 6 ⊢ ((◡𝐽‘𝐾) ∈ (𝑀..^𝐾) ↔ ((◡𝐽‘𝐾) ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ ∧ (◡𝐽‘𝐾) < 𝐾)) | |
27 | 21, 24, 25, 26 | syl3anbrc 1176 | . . . . 5 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (◡𝐽‘𝐾) ∈ (𝑀..^𝐾)) |
28 | 11, 13, 27 | rspcdva 2839 | . . . 4 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → (𝐽‘(◡𝐽‘𝐾)) = (◡𝐽‘𝐾)) |
29 | 8, 28 | eqtr3d 2205 | . . 3 ⊢ ((𝜑 ∧ (◡𝐽‘𝐾) < 𝐾) → 𝐾 = (◡𝐽‘𝐾)) |
30 | 2, 29 | mtand 660 | . 2 ⊢ (𝜑 → ¬ (◡𝐽‘𝐾) < 𝐾) |
31 | elfzelz 9981 | . . . 4 ⊢ ((◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (◡𝐽‘𝐾) ∈ ℤ) | |
32 | 18, 31 | syl 14 | . . 3 ⊢ (𝜑 → (◡𝐽‘𝐾) ∈ ℤ) |
33 | ztri3or 9255 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ (◡𝐽‘𝐾) ∈ ℤ) → (𝐾 < (◡𝐽‘𝐾) ∨ 𝐾 = (◡𝐽‘𝐾) ∨ (◡𝐽‘𝐾) < 𝐾)) | |
34 | 23, 32, 33 | syl2anc 409 | . 2 ⊢ (𝜑 → (𝐾 < (◡𝐽‘𝐾) ∨ 𝐾 = (◡𝐽‘𝐾) ∨ (◡𝐽‘𝐾) < 𝐾)) |
35 | 2, 30, 34 | ecase23d 1345 | 1 ⊢ (𝜑 → 𝐾 < (◡𝐽‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ w3o 972 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∀wral 2448 class class class wbr 3989 ◡ccnv 4610 ⟶wf 5194 –1-1-onto→wf1o 5197 ‘cfv 5198 (class class class)co 5853 < clt 7954 ℤcz 9212 ℤ≥cuz 9487 ...cfz 9965 ..^cfzo 10098 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-fzo 10099 |
This theorem is referenced by: seq3f1olemqsumkj 10454 |
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