![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > uzind4s | Structured version Visualization version 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. (Contributed by NM, 4-Nov-2005.) |
Ref | Expression |
---|---|
uzind4s.1 | ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑) |
uzind4s.2 | ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜑 → [(𝑘 + 1) / 𝑘]𝜑)) |
Ref | Expression |
---|---|
uzind4s | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑘]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3655 | . 2 ⊢ (𝑗 = 𝑀 → ([𝑗 / 𝑘]𝜑 ↔ [𝑀 / 𝑘]𝜑)) | |
2 | sbequ 2452 | . 2 ⊢ (𝑗 = 𝑚 → ([𝑗 / 𝑘]𝜑 ↔ [𝑚 / 𝑘]𝜑)) | |
3 | dfsbcq2 3655 | . 2 ⊢ (𝑗 = (𝑚 + 1) → ([𝑗 / 𝑘]𝜑 ↔ [(𝑚 + 1) / 𝑘]𝜑)) | |
4 | dfsbcq2 3655 | . 2 ⊢ (𝑗 = 𝑁 → ([𝑗 / 𝑘]𝜑 ↔ [𝑁 / 𝑘]𝜑)) | |
5 | uzind4s.1 | . 2 ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑) | |
6 | nfv 1957 | . . . 4 ⊢ Ⅎ𝑘 𝑚 ∈ (ℤ≥‘𝑀) | |
7 | nfs1v 2254 | . . . . 5 ⊢ Ⅎ𝑘[𝑚 / 𝑘]𝜑 | |
8 | nfsbc1v 3672 | . . . . 5 ⊢ Ⅎ𝑘[(𝑚 + 1) / 𝑘]𝜑 | |
9 | 7, 8 | nfim 1943 | . . . 4 ⊢ Ⅎ𝑘([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑) |
10 | 6, 9 | nfim 1943 | . . 3 ⊢ Ⅎ𝑘(𝑚 ∈ (ℤ≥‘𝑀) → ([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑)) |
11 | eleq1w 2842 | . . . 4 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ (ℤ≥‘𝑀) ↔ 𝑚 ∈ (ℤ≥‘𝑀))) | |
12 | sbequ12 2229 | . . . . 5 ⊢ (𝑘 = 𝑚 → (𝜑 ↔ [𝑚 / 𝑘]𝜑)) | |
13 | oveq1 6929 | . . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1)) | |
14 | 13 | sbceq1d 3657 | . . . . 5 ⊢ (𝑘 = 𝑚 → ([(𝑘 + 1) / 𝑘]𝜑 ↔ [(𝑚 + 1) / 𝑘]𝜑)) |
15 | 12, 14 | imbi12d 336 | . . . 4 ⊢ (𝑘 = 𝑚 → ((𝜑 → [(𝑘 + 1) / 𝑘]𝜑) ↔ ([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑))) |
16 | 11, 15 | imbi12d 336 | . . 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 2360 | . 2 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → ([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑)) |
19 | 1, 2, 3, 4, 5, 18 | uzind4 12052 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑘]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2011 ∈ wcel 2107 [wsbc 3652 ‘cfv 6135 (class class class)co 6922 1c1 10273 + caddc 10275 ℤcz 11728 ℤ≥cuz 11992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |