Theorem uzind4s 9925

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. (Contributed by NM, 4-Nov-2005.)

Hypotheses

Ref	Expression
uzind4s.1	⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑)
uzind4s.2	⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → [(𝑘 + 1) / 𝑘]𝜑))

Assertion

Ref	Expression
uzind4s	⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑘]𝜑)

Distinct variable group:   𝑘,𝑀

Allowed substitution hints:   𝜑(𝑘)   𝑁(𝑘)

Proof of Theorem uzind4s

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

Step	Hyp	Ref	Expression
1		dfsbcq2 3047	. 2 ⊢ (𝑗 = 𝑀 → ([𝑗 / 𝑘]𝜑 ↔ [𝑀 / 𝑘]𝜑))
2		sbequ 1889	. 2 ⊢ (𝑗 = 𝑚 → ([𝑗 / 𝑘]𝜑 ↔ [𝑚 / 𝑘]𝜑))
3		dfsbcq2 3047	. 2 ⊢ (𝑗 = (𝑚 + 1) → ([𝑗 / 𝑘]𝜑 ↔ [(𝑚 + 1) / 𝑘]𝜑))
4		dfsbcq2 3047	. 2 ⊢ (𝑗 = 𝑁 → ([𝑗 / 𝑘]𝜑 ↔ [𝑁 / 𝑘]𝜑))
5		uzind4s.1	. 2 ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑)
6		nfv 1577	. . . 4 ⊢ Ⅎ𝑘 𝑚 ∈ (ℤ≥‘𝑀)
7		nfs1v 1995	. . . . 5 ⊢ Ⅎ𝑘[𝑚 / 𝑘]𝜑
8		nfsbc1v 3063	. . . . 5 ⊢ Ⅎ𝑘[(𝑚 + 1) / 𝑘]𝜑
9	7, 8	nfim 1621	. . . 4 ⊢ Ⅎ𝑘([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑)
10	6, 9	nfim 1621	. . 3 ⊢ Ⅎ𝑘(𝑚 ∈ (ℤ≥‘𝑀) → ([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑))
11		eleq1 2297	. . . 4 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ (ℤ≥‘𝑀) ↔ 𝑚 ∈ (ℤ≥‘𝑀)))
12		sbequ12 1820	. . . . 5 ⊢ (𝑘 = 𝑚 → (𝜑 ↔ [𝑚 / 𝑘]𝜑))
13		oveq1 6059	. . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
14	13	sbceq1d 3049	. . . . 5 ⊢ (𝑘 = 𝑚 → ([(𝑘 + 1) / 𝑘]𝜑 ↔ [(𝑚 + 1) / 𝑘]𝜑))
15	12, 14	imbi12d 234	. . . 4 ⊢ (𝑘 = 𝑚 → ((𝜑 → [(𝑘 + 1) / 𝑘]𝜑) ↔ ([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑)))
16	11, 15	imbi12d 234	. . 3 ⊢ (𝑘 = 𝑚 → ((𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → [(𝑘 + 1) / 𝑘]𝜑)) ↔ (𝑚 ∈ (ℤ≥‘𝑀) → ([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑))))
17		uzind4s.2	. . 3 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → [(𝑘 + 1) / 𝑘]𝜑))
18	10, 16, 17	chvar 1806	. 2 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → ([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑))
19	1, 2, 3, 4, 5, 18	uzind4 9923	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑘]𝜑)

Syntax hints:   → wi 4  [wsb 1811   ∈ wcel 2205  [wsbc 3044  ‘cfv 5354  (class class class)co 6052  1c1 8130   + caddc 8132  ℤcz 9579  ℤ_≥cuz 9856

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857

This theorem is referenced by: (None)