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Mirrors > Home > ILE Home > Th. List > iseqf1olemnanb | GIF version |
Description: Lemma for seq3f1o 10277. (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 10260 | . . . 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 3484 | . . . 4 ⊢ (𝜑 → if(𝐴 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴)) = (𝐽‘𝐴)) |
9 | 6, 8 | eqtrd 2172 | . . 3 ⊢ (𝜑 → (𝑄‘𝐴) = (𝐽‘𝐴)) |
10 | iseqf1olemnab.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝑀...𝑁)) | |
11 | 2, 3, 10, 5 | iseqf1olemqval 10260 | . . . 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 3484 | . . . 4 ⊢ (𝜑 → if(𝐵 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽‘𝐵)) = (𝐽‘𝐵)) |
14 | 11, 13 | eqtrd 2172 | . . 3 ⊢ (𝜑 → (𝑄‘𝐵) = (𝐽‘𝐵)) |
15 | 1, 9, 14 | 3eqtr3d 2180 | . 2 ⊢ (𝜑 → (𝐽‘𝐴) = (𝐽‘𝐵)) |
16 | f1of1 5366 | . . . 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 5670 | . . 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 1214 | . 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 1331 ∈ wcel 1480 ifcif 3474 ↦ cmpt 3989 ◡ccnv 4538 –1-1→wf1 5120 –1-1-onto→wf1o 5122 ‘cfv 5123 (class class class)co 5774 1c1 7621 − cmin 7933 ...cfz 9790 |
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-dc 820 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-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 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-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 |
This theorem is referenced by: iseqf1olemmo 10265 |
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