Theorem uzind4s2 9621

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.)

Hypotheses

Ref	Expression
uzind4s2.1	⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)
uzind4s2.2	⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ([𝑘 / 𝑗]𝜑 → [(𝑘 + 1) / 𝑗]𝜑))

Assertion

Ref	Expression
uzind4s2	⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑗]𝜑)

Distinct variable groups:   𝑘,𝑀   𝜑,𝑘   𝑗,𝑘

Allowed substitution hints:   𝜑(𝑗)   𝑀(𝑗)   𝑁(𝑗,𝑘)

Proof of Theorem uzind4s2

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

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)

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)