![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem30 | Structured version Visualization version GIF version |
Description: This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, (𝐺‘𝑖) is used for p_(ti). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stoweidlem30.1 | ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} |
stoweidlem30.2 | ⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) |
stoweidlem30.3 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
stoweidlem30.4 | ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄) |
stoweidlem30.5 | ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
Ref | Expression |
---|---|
stoweidlem30 | ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2827 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝑇 ↔ 𝑆 ∈ 𝑇)) | |
2 | 1 | anbi2d 630 | . . . 4 ⊢ (𝑠 = 𝑆 → ((𝜑 ∧ 𝑠 ∈ 𝑇) ↔ (𝜑 ∧ 𝑆 ∈ 𝑇))) |
3 | fveq2 6907 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝑃‘𝑠) = (𝑃‘𝑆)) | |
4 | fveq2 6907 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝐺‘𝑖)‘𝑠) = ((𝐺‘𝑖)‘𝑆)) | |
5 | 4 | sumeq2sdv 15736 | . . . . . 6 ⊢ (𝑠 = 𝑆 → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠) = Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)) |
6 | 5 | oveq2d 7447 | . . . . 5 ⊢ (𝑠 = 𝑆 → ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠)) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆))) |
7 | 3, 6 | eqeq12d 2751 | . . . 4 ⊢ (𝑠 = 𝑆 → ((𝑃‘𝑠) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠)) ↔ (𝑃‘𝑆) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)))) |
8 | 2, 7 | imbi12d 344 | . . 3 ⊢ (𝑠 = 𝑆 → (((𝜑 ∧ 𝑠 ∈ 𝑇) → (𝑃‘𝑠) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠))) ↔ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆))))) |
9 | stoweidlem30.2 | . . . 4 ⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) | |
10 | fveq2 6907 | . . . . . 6 ⊢ (𝑡 = 𝑠 → ((𝐺‘𝑖)‘𝑡) = ((𝐺‘𝑖)‘𝑠)) | |
11 | 10 | sumeq2sdv 15736 | . . . . 5 ⊢ (𝑡 = 𝑠 → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡) = Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠)) |
12 | 11 | oveq2d 7447 | . . . 4 ⊢ (𝑡 = 𝑠 → ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠))) |
13 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → 𝑠 ∈ 𝑇) | |
14 | stoweidlem30.3 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
15 | 14 | nnrecred 12315 | . . . . . 6 ⊢ (𝜑 → (1 / 𝑀) ∈ ℝ) |
16 | 15 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → (1 / 𝑀) ∈ ℝ) |
17 | fzfid 14011 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → (1...𝑀) ∈ Fin) | |
18 | stoweidlem30.1 | . . . . . . . . 9 ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} | |
19 | stoweidlem30.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄) | |
20 | stoweidlem30.5 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) | |
21 | 18, 19, 20 | stoweidlem15 45971 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑠 ∈ 𝑇) → (((𝐺‘𝑖)‘𝑠) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑠) ∧ ((𝐺‘𝑖)‘𝑠) ≤ 1)) |
22 | 21 | simp1d 1141 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑠 ∈ 𝑇) → ((𝐺‘𝑖)‘𝑠) ∈ ℝ) |
23 | 22 | an32s 652 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑠) ∈ ℝ) |
24 | 17, 23 | fsumrecl 15767 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠) ∈ ℝ) |
25 | 16, 24 | remulcld 11289 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠)) ∈ ℝ) |
26 | 9, 12, 13, 25 | fvmptd3 7039 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → (𝑃‘𝑠) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑠))) |
27 | 8, 26 | vtoclg 3554 | . 2 ⊢ (𝑆 ∈ 𝑇 → ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)))) |
28 | 27 | anabsi7 671 | 1 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 ≤ cle 11294 / cdiv 11918 ℕcn 12264 ...cfz 13544 Σcsu 15719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 |
This theorem is referenced by: stoweidlem37 45993 stoweidlem38 45994 stoweidlem44 46000 |
Copyright terms: Public domain | W3C validator |