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Mirrors > Home > ILE Home > Th. List > uzind4i | GIF version |
Description: Induction on the upper integers that start at 𝑀. The first four give us the substitution instances we need, and the last two are the basis and the induction step. This is a stronger version of uzind4 9559 assuming that 𝜓 holds unconditionally. Notice that 𝑁 ∈ (ℤ≥‘𝑀) implies that the lower bound 𝑀 is an integer (𝑀 ∈ ℤ, see eluzel2 9504). (Contributed by NM, 4-Sep-2005.) (Revised by AV, 13-Jul-2022.) |
Ref | Expression |
---|---|
uzind4i.1 | ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) |
uzind4i.2 | ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) |
uzind4i.3 | ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) |
uzind4i.4 | ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) |
uzind4i.5 | ⊢ 𝜓 |
uzind4i.6 | ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
uzind4i | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzind4i.1 | . 2 ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) | |
2 | uzind4i.2 | . 2 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) | |
3 | uzind4i.3 | . 2 ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) | |
4 | uzind4i.4 | . 2 ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) | |
5 | uzind4i.5 | . . 3 ⊢ 𝜓 | |
6 | 5 | a1i 9 | . 2 ⊢ (𝑀 ∈ ℤ → 𝜓) |
7 | uzind4i.6 | . 2 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) | |
8 | 1, 2, 3, 4, 6, 7 | uzind4 9559 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2146 ‘cfv 5208 (class class class)co 5865 1c1 7787 + caddc 7789 ℤcz 9224 ℤ≥cuz 9499 |
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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-addass 7888 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-ltadd 7902 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-inn 8891 df-n0 9148 df-z 9225 df-uz 9500 |
This theorem is referenced by: rebtwn2zlemshrink 10222 |
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