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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 | ⊢ 0 = (𝐼 × {0}) |
rrx0el.p | ⊢ 𝑃 = (ℝ ↑𝑚 𝐼) |
Ref | Expression |
---|---|
rrx0el | ⊢ (𝐼 ∈ 𝑉 → 0 ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10357 | . . . . . 6 ⊢ 0 ∈ V | |
2 | 1 | fconst 6332 | . . . . 5 ⊢ (𝐼 × {0}):𝐼⟶{0} |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}):𝐼⟶{0}) |
4 | 0re 10365 | . . . . . 6 ⊢ 0 ∈ ℝ | |
5 | snssg 4536 | . . . . . . 7 ⊢ (0 ∈ ℝ → (0 ∈ ℝ ↔ {0} ⊆ ℝ)) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (0 ∈ ℝ ↔ {0} ⊆ ℝ) |
7 | 4, 6 | mpbi 222 | . . . . 5 ⊢ {0} ⊆ ℝ |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → {0} ⊆ ℝ) |
9 | 3, 8 | fssd 6296 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}):𝐼⟶ℝ) |
10 | reex 10350 | . . . . 5 ⊢ ℝ ∈ V | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝ ∈ V) |
12 | id 22 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉) | |
13 | 11, 12 | elmapd 8141 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((𝐼 × {0}) ∈ (ℝ ↑𝑚 𝐼) ↔ (𝐼 × {0}):𝐼⟶ℝ)) |
14 | 9, 13 | mpbird 249 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ (ℝ ↑𝑚 𝐼)) |
15 | rrx0el.0 | . 2 ⊢ 0 = (𝐼 × {0}) | |
16 | rrx0el.p | . 2 ⊢ 𝑃 = (ℝ ↑𝑚 𝐼) | |
17 | 14, 15, 16 | 3eltr4g 2923 | 1 ⊢ (𝐼 ∈ 𝑉 → 0 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ⊆ wss 3798 {csn 4399 × cxp 5344 ⟶wf 6123 (class class class)co 6910 ↑𝑚 cmap 8127 ℝcr 10258 0cc0 10259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-i2m1 10327 ax-rnegex 10330 ax-cnre 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 |
This theorem is referenced by: 2sphere0 43312 |
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