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Mirrors > Home > ILE Home > Th. List > iseqf1olemnanb | GIF version |
Description: Lemma for seq3f1o 10430. (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 10413 | . . . 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 3526 | . . . 4 ⊢ (𝜑 → if(𝐴 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴)) = (𝐽‘𝐴)) |
9 | 6, 8 | eqtrd 2197 | . . 3 ⊢ (𝜑 → (𝑄‘𝐴) = (𝐽‘𝐴)) |
10 | iseqf1olemnab.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝑀...𝑁)) | |
11 | 2, 3, 10, 5 | iseqf1olemqval 10413 | . . . 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 3526 | . . . 4 ⊢ (𝜑 → if(𝐵 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐵 = 𝐾, 𝐾, (𝐽‘(𝐵 − 1))), (𝐽‘𝐵)) = (𝐽‘𝐵)) |
14 | 11, 13 | eqtrd 2197 | . . 3 ⊢ (𝜑 → (𝑄‘𝐵) = (𝐽‘𝐵)) |
15 | 1, 9, 14 | 3eqtr3d 2205 | . 2 ⊢ (𝜑 → (𝐽‘𝐴) = (𝐽‘𝐵)) |
16 | f1of1 5426 | . . . 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 5732 | . . 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 1225 | . 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 1342 ∈ wcel 2135 ifcif 3516 ↦ cmpt 4038 ◡ccnv 4598 –1-1→wf1 5180 –1-1-onto→wf1o 5182 ‘cfv 5183 (class class class)co 5837 1c1 7746 − cmin 8061 ...cfz 9936 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-ltadd 7861 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-if 3517 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-inn 8850 df-n0 9107 df-z 9184 df-uz 9459 df-fz 9937 |
This theorem is referenced by: iseqf1olemmo 10418 |
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