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Mirrors > Home > ILE Home > Th. List > uzind4s2 | GIF version |
Description: Induction on the upper set of integers that starts at an integer 𝑀, using explicit substitution. The hypotheses are the basis and the induction step. Use this instead of uzind4s 9620 when 𝑗 and 𝑘 must be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM, 16-Nov-2005.) |
Ref | Expression |
---|---|
uzind4s2.1 | ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑) |
uzind4s2.2 | ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ([𝑘 / 𝑗]𝜑 → [(𝑘 + 1) / 𝑗]𝜑)) |
Ref | Expression |
---|---|
uzind4s2 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑗]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 2979 | . 2 ⊢ (𝑚 = 𝑀 → ([𝑚 / 𝑗]𝜑 ↔ [𝑀 / 𝑗]𝜑)) | |
2 | dfsbcq 2979 | . 2 ⊢ (𝑚 = 𝑛 → ([𝑚 / 𝑗]𝜑 ↔ [𝑛 / 𝑗]𝜑)) | |
3 | dfsbcq 2979 | . 2 ⊢ (𝑚 = (𝑛 + 1) → ([𝑚 / 𝑗]𝜑 ↔ [(𝑛 + 1) / 𝑗]𝜑)) | |
4 | dfsbcq 2979 | . 2 ⊢ (𝑚 = 𝑁 → ([𝑚 / 𝑗]𝜑 ↔ [𝑁 / 𝑗]𝜑)) | |
5 | uzind4s2.1 | . 2 ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑) | |
6 | dfsbcq 2979 | . . . 4 ⊢ (𝑘 = 𝑛 → ([𝑘 / 𝑗]𝜑 ↔ [𝑛 / 𝑗]𝜑)) | |
7 | oveq1 5903 | . . . . 5 ⊢ (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1)) | |
8 | 7 | sbceq1d 2982 | . . . 4 ⊢ (𝑘 = 𝑛 → ([(𝑘 + 1) / 𝑗]𝜑 ↔ [(𝑛 + 1) / 𝑗]𝜑)) |
9 | 6, 8 | imbi12d 234 | . . 3 ⊢ (𝑘 = 𝑛 → (([𝑘 / 𝑗]𝜑 → [(𝑘 + 1) / 𝑗]𝜑) ↔ ([𝑛 / 𝑗]𝜑 → [(𝑛 + 1) / 𝑗]𝜑))) |
10 | uzind4s2.2 | . . 3 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ([𝑘 / 𝑗]𝜑 → [(𝑘 + 1) / 𝑗]𝜑)) | |
11 | 9, 10 | vtoclga 2818 | . 2 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ([𝑛 / 𝑗]𝜑 → [(𝑛 + 1) / 𝑗]𝜑)) |
12 | 1, 2, 3, 4, 5, 11 | uzind4 9618 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑗]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 [wsbc 2977 ‘cfv 5235 (class class class)co 5896 1c1 7842 + caddc 7844 ℤcz 9283 ℤ≥cuz 9558 |
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 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-ltadd 7957 |
This theorem depends on definitions: df-bi 117 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-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-inn 8950 df-n0 9207 df-z 9284 df-uz 9559 |
This theorem is referenced by: (None) |
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