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Mirrors > Home > ILE Home > Th. List > iseqf1olemnanb | GIF version |
Description: Lemma for seq3f1o 10543. (Contributed by Jim Kingdon, 27-Aug-2022.) |
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)) |
iseqf1olemnab.b | ⊢ (𝜑 → 𝐵 ∈ (𝑀...𝑁)) |
iseqf1olemnab.eq | ⊢ (𝜑 → (𝑄‘𝐴) = (𝑄‘𝐵)) |
iseqf1olemnab.q | ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) |
iseqf1olemnanb.a | ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐾...(◡𝐽‘𝐾))) |
iseqf1olemnanb.b | ⊢ (𝜑 → ¬ 𝐵 ∈ (𝐾...(◡𝐽‘𝐾))) |
Ref | Expression |
---|---|
iseqf1olemnanb | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqf1olemnab.eq | . . 3 ⊢ (𝜑 → (𝑄‘𝐴) = (𝑄‘𝐵)) | |
2 | iseqf1olemqcl.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) | |
3 | iseqf1olemqcl.j | . . . . 5 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) | |
4 | iseqf1olemqcl.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁)) | |
5 | iseqf1olemnab.q | . . . . 5 ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) | |
6 | 2, 3, 4, 5 | iseqf1olemqval 10526 | . . . 4 ⊢ (𝜑 → (𝑄‘𝐴) = if(𝐴 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴))) |
7 | iseqf1olemnanb.a | . . . . 5 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐾...(◡𝐽‘𝐾))) | |
8 | 7 | iffalsed 3559 | . . . 4 ⊢ (𝜑 → if(𝐴 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴)) = (𝐽‘𝐴)) |
9 | 6, 8 | eqtrd 2222 | . . 3 ⊢ (𝜑 → (𝑄‘𝐴) = (𝐽‘𝐴)) |
10 | iseqf1olemnab.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝑀...𝑁)) | |
11 | 2, 3, 10, 5 | iseqf1olemqval 10526 | . . . 4 ⊢ (𝜑 → (𝑄‘𝐵) = if(𝐵 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽‘𝐵))) |
12 | iseqf1olemnanb.b | . . . . 5 ⊢ (𝜑 → ¬ 𝐵 ∈ (𝐾...(◡𝐽‘𝐾))) | |
13 | 12 | iffalsed 3559 | . . . 4 ⊢ (𝜑 → if(𝐵 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽‘𝐵)) = (𝐽‘𝐵)) |
14 | 11, 13 | eqtrd 2222 | . . 3 ⊢ (𝜑 → (𝑄‘𝐵) = (𝐽‘𝐵)) |
15 | 1, 9, 14 | 3eqtr3d 2230 | . 2 ⊢ (𝜑 → (𝐽‘𝐴) = (𝐽‘𝐵)) |
16 | f1of1 5482 | . . . 4 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1→(𝑀...𝑁)) | |
17 | 3, 16 | syl 14 | . . 3 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1→(𝑀...𝑁)) |
18 | f1veqaeq 5794 | . . 3 ⊢ ((𝐽:(𝑀...𝑁)–1-1→(𝑀...𝑁) ∧ (𝐴 ∈ (𝑀...𝑁) ∧ 𝐵 ∈ (𝑀...𝑁))) → ((𝐽‘𝐴) = (𝐽‘𝐵) → 𝐴 = 𝐵)) | |
19 | 17, 4, 10, 18 | syl12anc 1247 | . 2 ⊢ (𝜑 → ((𝐽‘𝐴) = (𝐽‘𝐵) → 𝐴 = 𝐵)) |
20 | 15, 19 | mpd 13 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1364 ∈ wcel 2160 ifcif 3549 ↦ cmpt 4082 ◡ccnv 4646 –1-1→wf1 5235 –1-1-onto→wf1o 5237 ‘cfv 5238 (class class class)co 5900 1c1 7847 − cmin 8163 ...cfz 10044 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-addcom 7946 ax-addass 7948 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-0id 7954 ax-rnegex 7955 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-ltadd 7962 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-inn 8955 df-n0 9212 df-z 9289 df-uz 9564 df-fz 10045 |
This theorem is referenced by: iseqf1olemmo 10531 |
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