![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rrx0el | Structured version Visualization version GIF version |
Description: The zero ("origin") in a generalized real Euclidean space is an element of its base set. (Contributed by AV, 11-Feb-2023.) |
Ref | Expression |
---|---|
rrx0el.0 | β’ 0 = (πΌ Γ {0}) |
rrx0el.p | β’ π = (β βm πΌ) |
Ref | Expression |
---|---|
rrx0el | β’ (πΌ β π β 0 β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 11150 | . . . . . 6 β’ 0 β V | |
2 | 1 | fconst 6729 | . . . . 5 β’ (πΌ Γ {0}):πΌβΆ{0} |
3 | 2 | a1i 11 | . . . 4 β’ (πΌ β π β (πΌ Γ {0}):πΌβΆ{0}) |
4 | 0re 11158 | . . . . . 6 β’ 0 β β | |
5 | snssg 4745 | . . . . . . 7 β’ (0 β β β (0 β β β {0} β β)) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 β’ (0 β β β {0} β β) |
7 | 4, 6 | mpbi 229 | . . . . 5 β’ {0} β β |
8 | 7 | a1i 11 | . . . 4 β’ (πΌ β π β {0} β β) |
9 | 3, 8 | fssd 6687 | . . 3 β’ (πΌ β π β (πΌ Γ {0}):πΌβΆβ) |
10 | reex 11143 | . . . . 5 β’ β β V | |
11 | 10 | a1i 11 | . . . 4 β’ (πΌ β π β β β V) |
12 | id 22 | . . . 4 β’ (πΌ β π β πΌ β π) | |
13 | 11, 12 | elmapd 8780 | . . 3 β’ (πΌ β π β ((πΌ Γ {0}) β (β βm πΌ) β (πΌ Γ {0}):πΌβΆβ)) |
14 | 9, 13 | mpbird 257 | . 2 β’ (πΌ β π β (πΌ Γ {0}) β (β βm πΌ)) |
15 | rrx0el.0 | . 2 β’ 0 = (πΌ Γ {0}) | |
16 | rrx0el.p | . 2 β’ π = (β βm πΌ) | |
17 | 14, 15, 16 | 3eltr4g 2855 | 1 β’ (πΌ β π β 0 β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 Vcvv 3446 β wss 3911 {csn 4587 Γ cxp 5632 βΆwf 6493 (class class class)co 7358 βm cmap 8766 βcr 11051 0cc0 11052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-i2m1 11120 ax-rnegex 11123 ax-cnre 11125 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8768 |
This theorem is referenced by: ehl2eudisval0 46818 2sphere0 46843 |
Copyright terms: Public domain | W3C validator |