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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > finfdm2 | Structured version Visualization version GIF version |
Description: The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 46774. Note that this definition of the inf function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fifth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 1-Feb-2025.) |
Ref | Expression |
---|---|
finfdm2.1 | ⊢ Ⅎ𝑛𝜑 |
finfdm2.2 | ⊢ Ⅎ𝑥𝜑 |
finfdm2.3 | ⊢ Ⅎ𝑚𝜑 |
finfdm2.4 | ⊢ Ⅎ𝑥𝐹 |
finfdm2.5 | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ*) |
finfdm2.6 | ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} |
finfdm2.7 | ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
finfdm2.8 | ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})) |
Ref | Expression |
---|---|
finfdm2 | ⊢ (𝜑 → dom 𝐺 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | finfdm2.2 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | finfdm2.6 | . . . 4 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} | |
3 | nfrab1 3454 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} | |
4 | 2, 3 | nfcxfr 2901 | . . 3 ⊢ Ⅎ𝑥𝐷 |
5 | finfdm2.7 | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) | |
6 | ltso 11339 | . . . . 5 ⊢ < Or ℝ | |
7 | 6 | infex 9531 | . . . 4 ⊢ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V |
8 | 7 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V) |
9 | 1, 4, 5, 8 | dmmptdff 45166 | . 2 ⊢ (𝜑 → dom 𝐺 = 𝐷) |
10 | finfdm2.1 | . . 3 ⊢ Ⅎ𝑛𝜑 | |
11 | finfdm2.3 | . . 3 ⊢ Ⅎ𝑚𝜑 | |
12 | finfdm2.4 | . . 3 ⊢ Ⅎ𝑥𝐹 | |
13 | finfdm2.5 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ*) | |
14 | finfdm2.8 | . . 3 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})) | |
15 | 10, 1, 11, 12, 13, 2, 14 | finfdm 46802 | . 2 ⊢ (𝜑 → 𝐷 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) |
16 | 9, 15 | eqtrd 2775 | 1 ⊢ (𝜑 → dom 𝐺 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ∀wral 3059 ∃wrex 3068 {crab 3433 Vcvv 3478 ∪ ciun 4996 ∩ ciin 4997 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 ⟶wf 6559 ‘cfv 6563 infcinf 9479 ℝcr 11152 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 -cneg 11491 ℕcn 12264 |
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-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-iun 4998 df-iin 4999 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-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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 |
This theorem is referenced by: smfinfdmmbllem 46804 |
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