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Mirrors > Home > MPE Home > Th. List > fzindd | Structured version Visualization version GIF version |
Description: Induction on the integers from M to N inclusive, a deduction version. (Contributed by metakunt, 12-May-2024.) |
Ref | Expression |
---|---|
fzindd.1 | ⊢ (𝑥 = 𝑀 → (𝜓 ↔ 𝜒)) |
fzindd.2 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) |
fzindd.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) |
fzindd.4 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) |
fzindd.5 | ⊢ (𝜑 → 𝜒) |
fzindd.6 | ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁) ∧ 𝜃) → 𝜏) |
fzindd.7 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
fzindd.8 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
fzindd.9 | ⊢ (𝜑 → 𝑀 ≤ 𝑁) |
Ref | Expression |
---|---|
fzindd | ⊢ ((𝜑 ∧ (𝐴 ∈ ℤ ∧ 𝑀 ≤ 𝐴 ∧ 𝐴 ≤ 𝑁)) → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzindd.7 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | fzindd.8 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | 1, 2 | jca 513 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
4 | fzindd.1 | . . . . . 6 ⊢ (𝑥 = 𝑀 → (𝜓 ↔ 𝜒)) | |
5 | 4 | imbi2d 341 | . . . . 5 ⊢ (𝑥 = 𝑀 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
6 | fzindd.2 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) | |
7 | 6 | imbi2d 341 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜃))) |
8 | fzindd.3 | . . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝜓 ↔ 𝜏)) | |
9 | 8 | imbi2d 341 | . . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜏))) |
10 | fzindd.4 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) | |
11 | 10 | imbi2d 341 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜂))) |
12 | fzindd.5 | . . . . . 6 ⊢ (𝜑 → 𝜒) | |
13 | 12 | a1i 11 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝜑 → 𝜒)) |
14 | fzindd.6 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁) ∧ 𝜃) → 𝜏) | |
15 | 14 | 3expa 1119 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁)) ∧ 𝜃) → 𝜏) |
16 | 15 | ex 414 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (𝜃 → 𝜏)) |
17 | 16 | expcom 415 | . . . . . . 7 ⊢ ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁) → (𝜑 → (𝜃 → 𝜏))) |
18 | 17 | a2d 29 | . . . . . 6 ⊢ ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁) → ((𝜑 → 𝜃) → (𝜑 → 𝜏))) |
19 | 18 | adantl 483 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → ((𝜑 → 𝜃) → (𝜑 → 𝜏))) |
20 | 5, 7, 9, 11, 13, 19 | fzind 12602 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐴 ∈ ℤ ∧ 𝑀 ≤ 𝐴 ∧ 𝐴 ≤ 𝑁)) → (𝜑 → 𝜂)) |
21 | 3, 20 | sylan 581 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ ℤ ∧ 𝑀 ≤ 𝐴 ∧ 𝐴 ≤ 𝑁)) → (𝜑 → 𝜂)) |
22 | 21 | imp 408 | . 2 ⊢ (((𝜑 ∧ (𝐴 ∈ ℤ ∧ 𝑀 ≤ 𝐴 ∧ 𝐴 ≤ 𝑁)) ∧ 𝜑) → 𝜂) |
23 | 22 | anabss1 665 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ ℤ ∧ 𝑀 ≤ 𝐴 ∧ 𝐴 ≤ 𝑁)) → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 1c1 11053 + caddc 11055 < clt 11190 ≤ cle 11191 ℤcz 12500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-n0 12415 df-z 12501 |
This theorem is referenced by: lcmineqlem13 40501 natglobalincr 45123 |
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