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Mirrors > Home > ILE Home > Th. List > uzind4s | 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 2818 | . 2 ⊢ (𝑗 = 𝑀 → ([𝑗 / 𝑘]𝜑 ↔ [𝑀 / 𝑘]𝜑)) | |
2 | sbequ 1761 | . 2 ⊢ (𝑗 = 𝑚 → ([𝑗 / 𝑘]𝜑 ↔ [𝑚 / 𝑘]𝜑)) | |
3 | dfsbcq2 2818 | . 2 ⊢ (𝑗 = (𝑚 + 1) → ([𝑗 / 𝑘]𝜑 ↔ [(𝑚 + 1) / 𝑘]𝜑)) | |
4 | dfsbcq2 2818 | . 2 ⊢ (𝑗 = 𝑁 → ([𝑗 / 𝑘]𝜑 ↔ [𝑁 / 𝑘]𝜑)) | |
5 | uzind4s.1 | . 2 ⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑) | |
6 | nfv 1461 | . . . 4 ⊢ Ⅎ𝑘 𝑚 ∈ (ℤ≥‘𝑀) | |
7 | nfs1v 1856 | . . . . 5 ⊢ Ⅎ𝑘[𝑚 / 𝑘]𝜑 | |
8 | nfsbc1v 2833 | . . . . 5 ⊢ Ⅎ𝑘[(𝑚 + 1) / 𝑘]𝜑 | |
9 | 7, 8 | nfim 1504 | . . . 4 ⊢ Ⅎ𝑘([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑) |
10 | 6, 9 | nfim 1504 | . . 3 ⊢ Ⅎ𝑘(𝑚 ∈ (ℤ≥‘𝑀) → ([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑)) |
11 | eleq1 2141 | . . . 4 ⊢ (𝑘 = 𝑚 → (𝑘 ∈ (ℤ≥‘𝑀) ↔ 𝑚 ∈ (ℤ≥‘𝑀))) | |
12 | sbequ12 1694 | . . . . 5 ⊢ (𝑘 = 𝑚 → (𝜑 ↔ [𝑚 / 𝑘]𝜑)) | |
13 | oveq1 5539 | . . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1)) | |
14 | 13 | sbceq1d 2820 | . . . . 5 ⊢ (𝑘 = 𝑚 → ([(𝑘 + 1) / 𝑘]𝜑 ↔ [(𝑚 + 1) / 𝑘]𝜑)) |
15 | 12, 14 | imbi12d 232 | . . . 4 ⊢ (𝑘 = 𝑚 → ((𝜑 → [(𝑘 + 1) / 𝑘]𝜑) ↔ ([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑))) |
16 | 11, 15 | imbi12d 232 | . . 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 1680 | . 2 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → ([𝑚 / 𝑘]𝜑 → [(𝑚 + 1) / 𝑘]𝜑)) |
19 | 1, 2, 3, 4, 5, 18 | uzind4 8676 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → [𝑁 / 𝑘]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 [wsb 1685 [wsbc 2815 ‘cfv 4922 (class class class)co 5532 1c1 6982 + caddc 6984 ℤcz 8351 ℤ≥cuz 8619 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 |
This theorem is referenced by: (None) |
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