Theorem uzind4s2 8988

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 8987 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 2830	. 2 ⊢ (𝑚 = 𝑀 → ([𝑚 / 𝑗]𝜑 ↔ [𝑀 / 𝑗]𝜑))
2		dfsbcq 2830	. 2 ⊢ (𝑚 = 𝑛 → ([𝑚 / 𝑗]𝜑 ↔ [𝑛 / 𝑗]𝜑))
3		dfsbcq 2830	. 2 ⊢ (𝑚 = (𝑛 + 1) → ([𝑚 / 𝑗]𝜑 ↔ [(𝑛 + 1) / 𝑗]𝜑))
4		dfsbcq 2830	. 2 ⊢ (𝑚 = 𝑁 → ([𝑚 / 𝑗]𝜑 ↔ [𝑁 / 𝑗]𝜑))
5		uzind4s2.1	. 2 ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)
6		dfsbcq 2830	. . . 4 ⊢ (𝑘 = 𝑛 → ([𝑘 / 𝑗]𝜑 ↔ [𝑛 / 𝑗]𝜑))
7		oveq1 5601	. . . . 5 ⊢ (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
8	7	sbceq1d 2833	. . . 4 ⊢ (𝑘 = 𝑛 → ([(𝑘 + 1) / 𝑗]𝜑 ↔ [(𝑛 + 1) / 𝑗]𝜑))
9	6, 8	imbi12d 232	. . 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 2677	. 2 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ([𝑛 / 𝑗]𝜑 → [(𝑛 + 1) / 𝑗]𝜑))
12	1, 2, 3, 4, 5, 11	uzind4 8985	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑗]𝜑)

Syntax hints:   → wi 4   ∈ wcel 1436  [wsbc 2828  ‘cfv 4972  (class class class)co 5594  1c1 7272   + caddc 7274  ℤcz 8660  ℤ_≥cuz 8928

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-ltadd 7382

This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-inn 8335  df-n0 8584  df-z 8661  df-uz 8929

This theorem is referenced by: (None)