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Mirrors > Home > MPE Home > Th. List > mapdom3 | Structured version Visualization version GIF version |
Description: Set exponentiation dominates the mantissa. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
mapdom3 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑𝑚 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3931 | . . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐵) | |
2 | oveq1 6657 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑦 ↑𝑚 {𝑥}) = (𝐴 ↑𝑚 {𝑥})) | |
3 | id 22 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴) | |
4 | 2, 3 | breq12d 4666 | . . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑦 ↑𝑚 {𝑥}) ≈ 𝑦 ↔ (𝐴 ↑𝑚 {𝑥}) ≈ 𝐴)) |
5 | vex 3203 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
6 | vex 3203 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
7 | 5, 6 | mapsnen 8035 | . . . . . . . . 9 ⊢ (𝑦 ↑𝑚 {𝑥}) ≈ 𝑦 |
8 | 4, 7 | vtoclg 3266 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 {𝑥}) ≈ 𝐴) |
9 | 8 | 3ad2ant1 1082 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 ↑𝑚 {𝑥}) ≈ 𝐴) |
10 | 9 | ensymd 8007 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐴 ≈ (𝐴 ↑𝑚 {𝑥})) |
11 | simp2 1062 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ 𝑊) | |
12 | simp3 1063 | . . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
13 | 12 | snssd 4340 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → {𝑥} ⊆ 𝐵) |
14 | ssdomg 8001 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → ({𝑥} ⊆ 𝐵 → {𝑥} ≼ 𝐵)) | |
15 | 11, 13, 14 | sylc 65 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → {𝑥} ≼ 𝐵) |
16 | 6 | snnz 4309 | . . . . . . . 8 ⊢ {𝑥} ≠ ∅ |
17 | simpl 473 | . . . . . . . . 9 ⊢ (({𝑥} = ∅ ∧ 𝐴 = ∅) → {𝑥} = ∅) | |
18 | 17 | necon3ai 2819 | . . . . . . . 8 ⊢ ({𝑥} ≠ ∅ → ¬ ({𝑥} = ∅ ∧ 𝐴 = ∅)) |
19 | 16, 18 | ax-mp 5 | . . . . . . 7 ⊢ ¬ ({𝑥} = ∅ ∧ 𝐴 = ∅) |
20 | mapdom2 8131 | . . . . . . 7 ⊢ (({𝑥} ≼ 𝐵 ∧ ¬ ({𝑥} = ∅ ∧ 𝐴 = ∅)) → (𝐴 ↑𝑚 {𝑥}) ≼ (𝐴 ↑𝑚 𝐵)) | |
21 | 15, 19, 20 | sylancl 694 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 ↑𝑚 {𝑥}) ≼ (𝐴 ↑𝑚 𝐵)) |
22 | endomtr 8014 | . . . . . 6 ⊢ ((𝐴 ≈ (𝐴 ↑𝑚 {𝑥}) ∧ (𝐴 ↑𝑚 {𝑥}) ≼ (𝐴 ↑𝑚 𝐵)) → 𝐴 ≼ (𝐴 ↑𝑚 𝐵)) | |
23 | 10, 21, 22 | syl2anc 693 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐴 ≼ (𝐴 ↑𝑚 𝐵)) |
24 | 23 | 3expia 1267 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐵 → 𝐴 ≼ (𝐴 ↑𝑚 𝐵))) |
25 | 24 | exlimdv 1861 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 ∈ 𝐵 → 𝐴 ≼ (𝐴 ↑𝑚 𝐵))) |
26 | 1, 25 | syl5bi 232 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ≠ ∅ → 𝐴 ≼ (𝐴 ↑𝑚 𝐵))) |
27 | 26 | 3impia 1261 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑𝑚 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 ∅c0 3915 {csn 4177 class class class wbr 4653 (class class class)co 6650 ↑𝑚 cmap 7857 ≈ cen 7952 ≼ cdom 7953 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 |
This theorem is referenced by: infmap2 9040 |
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