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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rexanuz3 | Structured version Visualization version GIF version |
Description: Combine two different upper integer properties into one, for a single integer. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
rexanuz3.1 | ⊢ Ⅎ𝑗𝜑 |
rexanuz3.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
rexanuz3.3 | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜒) |
rexanuz3.4 | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) |
rexanuz3.5 | ⊢ (𝑘 = 𝑗 → (𝜒 ↔ 𝜃)) |
rexanuz3.6 | ⊢ (𝑘 = 𝑗 → (𝜓 ↔ 𝜏)) |
Ref | Expression |
---|---|
rexanuz3 | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝜃 ∧ 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexanuz3.3 | . . . 4 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜒) | |
2 | rexanuz3.4 | . . . 4 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓) | |
3 | 1, 2 | jca 509 | . . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜒 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)) |
4 | rexanuz3.2 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 4 | rexanuz2 14466 | . . 3 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜒 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)) |
6 | 3, 5 | sylibr 226 | . 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) |
7 | rexanuz3.1 | . . 3 ⊢ Ⅎ𝑗𝜑 | |
8 | 4 | eleq2i 2898 | . . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀)) |
9 | 8 | biimpi 208 | . . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀)) |
10 | eluzelz 11978 | . . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ) | |
11 | uzid 11983 | . . . . . . . . 9 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ≥‘𝑗)) | |
12 | 9, 10, 11 | 3syl 18 | . . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑗)) |
13 | 12 | adantr 474 | . . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) → 𝑗 ∈ (ℤ≥‘𝑗)) |
14 | simpr 479 | . . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) | |
15 | rexanuz3.5 | . . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝜒 ↔ 𝜃)) | |
16 | rexanuz3.6 | . . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝜓 ↔ 𝜏)) | |
17 | 15, 16 | anbi12d 626 | . . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏))) |
18 | 17 | rspcva 3524 | . . . . . . 7 ⊢ ((𝑗 ∈ (ℤ≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) → (𝜃 ∧ 𝜏)) |
19 | 13, 14, 18 | syl2anc 581 | . . . . . 6 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) → (𝜃 ∧ 𝜏)) |
20 | 19 | adantll 707 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) → (𝜃 ∧ 𝜏)) |
21 | 20 | ex 403 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓) → (𝜃 ∧ 𝜏))) |
22 | 21 | ex 403 | . . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓) → (𝜃 ∧ 𝜏)))) |
23 | 7, 22 | reximdai 3220 | . 2 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓) → ∃𝑗 ∈ 𝑍 (𝜃 ∧ 𝜏))) |
24 | 6, 23 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝜃 ∧ 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 ∀wral 3117 ∃wrex 3118 ‘cfv 6123 ℤcz 11704 ℤ≥cuz 11968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-pre-lttri 10326 ax-pre-lttrn 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-neg 10588 df-z 11705 df-uz 11969 |
This theorem is referenced by: smflimlem4 41776 |
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