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Mirrors > Home > MPE Home > Th. List > coe1mul2lem1 | Structured version Visualization version GIF version |
Description: An equivalence for coe1mul2 19639. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
Ref | Expression |
---|---|
coe1mul2lem1 | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝑋 ∘𝑟 ≤ (1𝑜 × {𝐴}) ↔ (𝑋‘∅) ∈ (0...𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 7567 | . . . 4 ⊢ 1𝑜 ∈ On | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → 1𝑜 ∈ On) |
3 | fvexd 6203 | . . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) ∧ 𝑎 ∈ 1𝑜) → (𝑋‘∅) ∈ V) | |
4 | simpll 790 | . . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) ∧ 𝑎 ∈ 1𝑜) → 𝐴 ∈ ℕ0) | |
5 | df1o2 7572 | . . . . . 6 ⊢ 1𝑜 = {∅} | |
6 | nn0ex 11298 | . . . . . 6 ⊢ ℕ0 ∈ V | |
7 | 0ex 4790 | . . . . . 6 ⊢ ∅ ∈ V | |
8 | 5, 6, 7 | mapsnconst 7903 | . . . . 5 ⊢ (𝑋 ∈ (ℕ0 ↑𝑚 1𝑜) → 𝑋 = (1𝑜 × {(𝑋‘∅)})) |
9 | 8 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑋 = (1𝑜 × {(𝑋‘∅)})) |
10 | fconstmpt 5163 | . . . 4 ⊢ (1𝑜 × {(𝑋‘∅)}) = (𝑎 ∈ 1𝑜 ↦ (𝑋‘∅)) | |
11 | 9, 10 | syl6eq 2672 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → 𝑋 = (𝑎 ∈ 1𝑜 ↦ (𝑋‘∅))) |
12 | fconstmpt 5163 | . . . 4 ⊢ (1𝑜 × {𝐴}) = (𝑎 ∈ 1𝑜 ↦ 𝐴) | |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → (1𝑜 × {𝐴}) = (𝑎 ∈ 1𝑜 ↦ 𝐴)) |
14 | 2, 3, 4, 11, 13 | ofrfval2 6915 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝑋 ∘𝑟 ≤ (1𝑜 × {𝐴}) ↔ ∀𝑎 ∈ 1𝑜 (𝑋‘∅) ≤ 𝐴)) |
15 | 1n0 7575 | . . 3 ⊢ 1𝑜 ≠ ∅ | |
16 | r19.3rzv 4064 | . . 3 ⊢ (1𝑜 ≠ ∅ → ((𝑋‘∅) ≤ 𝐴 ↔ ∀𝑎 ∈ 1𝑜 (𝑋‘∅) ≤ 𝐴)) | |
17 | 15, 16 | mp1i 13 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝑋‘∅) ≤ 𝐴 ↔ ∀𝑎 ∈ 1𝑜 (𝑋‘∅) ≤ 𝐴)) |
18 | elmapi 7879 | . . . . . 6 ⊢ (𝑋 ∈ (ℕ0 ↑𝑚 1𝑜) → 𝑋:1𝑜⟶ℕ0) | |
19 | 0lt1o 7584 | . . . . . 6 ⊢ ∅ ∈ 1𝑜 | |
20 | ffvelrn 6357 | . . . . . 6 ⊢ ((𝑋:1𝑜⟶ℕ0 ∧ ∅ ∈ 1𝑜) → (𝑋‘∅) ∈ ℕ0) | |
21 | 18, 19, 20 | sylancl 694 | . . . . 5 ⊢ (𝑋 ∈ (ℕ0 ↑𝑚 1𝑜) → (𝑋‘∅) ∈ ℕ0) |
22 | 21 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝑋‘∅) ∈ ℕ0) |
23 | 22 | biantrurd 529 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝑋‘∅) ≤ 𝐴 ↔ ((𝑋‘∅) ∈ ℕ0 ∧ (𝑋‘∅) ≤ 𝐴))) |
24 | fznn0 12432 | . . . 4 ⊢ (𝐴 ∈ ℕ0 → ((𝑋‘∅) ∈ (0...𝐴) ↔ ((𝑋‘∅) ∈ ℕ0 ∧ (𝑋‘∅) ≤ 𝐴))) | |
25 | 24 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝑋‘∅) ∈ (0...𝐴) ↔ ((𝑋‘∅) ∈ ℕ0 ∧ (𝑋‘∅) ≤ 𝐴))) |
26 | 23, 25 | bitr4d 271 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → ((𝑋‘∅) ≤ 𝐴 ↔ (𝑋‘∅) ∈ (0...𝐴))) |
27 | 14, 17, 26 | 3bitr2d 296 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝑋 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝑋 ∘𝑟 ≤ (1𝑜 × {𝐴}) ↔ (𝑋‘∅) ∈ (0...𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∅c0 3915 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 Oncon0 5723 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑟 cofr 6896 1𝑜c1o 7553 ↑𝑚 cmap 7857 0cc0 9936 ≤ cle 10075 ℕ0cn0 11292 ...cfz 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-fz 12327 |
This theorem is referenced by: coe1mul2lem2 19638 coe1mul2 19639 |
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