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Mirrors > Home > MPE Home > Th. List > 1arithlem1 | Structured version Visualization version GIF version |
Description: Lemma for 1arith 16356. (Contributed by Mario Carneiro, 30-May-2014.) |
Ref | Expression |
---|---|
1arith.1 | ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛))) |
Ref | Expression |
---|---|
1arithlem1 | ⊢ (𝑁 ∈ ℕ → (𝑀‘𝑁) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7172 | . . 3 ⊢ (𝑛 = 𝑁 → (𝑝 pCnt 𝑛) = (𝑝 pCnt 𝑁)) | |
2 | 1 | mpteq2dv 5123 | . 2 ⊢ (𝑛 = 𝑁 → (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛)) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑁))) |
3 | 1arith.1 | . 2 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑛))) | |
4 | prmex 16111 | . . 3 ⊢ ℙ ∈ V | |
5 | 4 | mptex 6990 | . 2 ⊢ (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑁)) ∈ V |
6 | 2, 3, 5 | fvmpt 6769 | 1 ⊢ (𝑁 ∈ ℕ → (𝑀‘𝑁) = (𝑝 ∈ ℙ ↦ (𝑝 pCnt 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ↦ cmpt 5107 ‘cfv 6333 (class class class)co 7164 ℕcn 11709 ℙcprime 16105 pCnt cpc 16266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-1cn 10666 ax-addcl 10668 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-nn 11710 df-prm 16106 |
This theorem is referenced by: 1arithlem2 16353 1arithlem3 16354 sqff1o 25911 |
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