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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 511 | . . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜒 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)) |
4 | rexanuz3.2 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 4 | rexanuz2 15314 | . . 3 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜒 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)) |
6 | 3, 5 | sylibr 233 | . 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) |
7 | rexanuz3.1 | . . 3 ⊢ Ⅎ𝑗𝜑 | |
8 | 4 | eleq2i 2820 | . . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀)) |
9 | 8 | biimpi 215 | . . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀)) |
10 | eluzelz 12848 | . . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ) | |
11 | uzid 12853 | . . . . . . . . 9 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ≥‘𝑗)) | |
12 | 9, 10, 11 | 3syl 18 | . . . . . . . 8 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑗)) |
13 | 12 | adantr 480 | . . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) → 𝑗 ∈ (ℤ≥‘𝑗)) |
14 | simpr 484 | . . . . . . 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 630 | . . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜏))) |
18 | 17 | rspcva 3605 | . . . . . . 7 ⊢ ((𝑗 ∈ (ℤ≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) → (𝜃 ∧ 𝜏)) |
19 | 13, 14, 18 | syl2anc 583 | . . . . . 6 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) → (𝜃 ∧ 𝜏)) |
20 | 19 | adantll 713 | . . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓)) → (𝜃 ∧ 𝜏)) |
21 | 20 | ex 412 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓) → (𝜃 ∧ 𝜏))) |
22 | 21 | ex 412 | . . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → (∀𝑘 ∈ (ℤ≥‘𝑗)(𝜒 ∧ 𝜓) → (𝜃 ∧ 𝜏)))) |
23 | 7, 22 | reximdai 3253 | . 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 205 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 ‘cfv 6542 ℤcz 12574 ℤ≥cuz 12838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-pre-lttri 11198 ax-pre-lttrn 11199 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-neg 11463 df-z 12575 df-uz 12839 |
This theorem is referenced by: smflimlem4 46075 |
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