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Mirrors > Home > ILE Home > Th. List > iseqf1olemnanb | GIF version |
Description: Lemma for seq3f1o 9987. (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 9970 | . . . 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 3407 | . . . 4 ⊢ (𝜑 → if(𝐴 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴)) = (𝐽‘𝐴)) |
9 | 6, 8 | eqtrd 2121 | . . 3 ⊢ (𝜑 → (𝑄‘𝐴) = (𝐽‘𝐴)) |
10 | iseqf1olemnab.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝑀...𝑁)) | |
11 | 2, 3, 10, 5 | iseqf1olemqval 9970 | . . . 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 3407 | . . . 4 ⊢ (𝜑 → if(𝐵 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽‘𝐵)) = (𝐽‘𝐵)) |
14 | 11, 13 | eqtrd 2121 | . . 3 ⊢ (𝜑 → (𝑄‘𝐵) = (𝐽‘𝐵)) |
15 | 1, 9, 14 | 3eqtr3d 2129 | . 2 ⊢ (𝜑 → (𝐽‘𝐴) = (𝐽‘𝐵)) |
16 | f1of1 5265 | . . . 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 5562 | . . 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 1173 | . 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 1290 ∈ wcel 1439 ifcif 3397 ↦ cmpt 3905 ◡ccnv 4450 –1-1→wf1 5025 –1-1-onto→wf1o 5027 ‘cfv 5028 (class class class)co 5666 1c1 7405 − cmin 7707 ...cfz 9478 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-addcom 7499 ax-addass 7501 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-0id 7507 ax-rnegex 7508 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-ltadd 7515 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-inn 8477 df-n0 8728 df-z 8805 df-uz 9074 df-fz 9479 |
This theorem is referenced by: iseqf1olemmo 9975 |
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