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Theorem imasdsfn 17528

Description: The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015.) (Proof shortened by AV, 6-Oct-2020.)

**Hypotheses**
Ref	Expression
imasbas.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imasbas.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasbas.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasds.e	⊢ 𝐸 = (dist‘𝑅)
imasds.d	⊢ 𝐷 = (dist‘𝑈)

**Assertion**
Ref	Expression
imasdsfn	⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))

Proof of Theorem imasdsfn Dummy variables 𝑔 ℎ 𝑖 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) ∣ ((𝐹‘(1^st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2^nd ‘(ℎ‘𝑖))) = (𝐹‘(1^st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < )) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) ∣ ((𝐹‘(1^st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2^nd ‘(ℎ‘𝑖))) = (𝐹‘(1^st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < ))
2		xrltso 13157	. . . 4 ⊢ < Or ℝ^*
3	2	infex 9507	. . 3 ⊢ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) ∣ ((𝐹‘(1^st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2^nd ‘(ℎ‘𝑖))) = (𝐹‘(1^st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < ) ∈ V
4	1, 3	fnmpoi 8069	. 2 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) ∣ ((𝐹‘(1^st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2^nd ‘(ℎ‘𝑖))) = (𝐹‘(1^st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < )) Fn (𝐵 × 𝐵)
5		imasbas.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
6		imasbas.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
7		imasbas.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
8		imasbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
9		imasds.e	. . . 4 ⊢ 𝐸 = (dist‘𝑅)
10		imasds.d	. . . 4 ⊢ 𝐷 = (dist‘𝑈)
11	5, 6, 7, 8, 9, 10	imasds 17527	. . 3 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) ∣ ((𝐹‘(1^st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2^nd ‘(ℎ‘𝑖))) = (𝐹‘(1^st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < )))
12	11	fneq1d 6631	. 2 ⊢ (𝜑 → (𝐷 Fn (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) ∣ ((𝐹‘(1^st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2^nd ‘(ℎ‘𝑖))) = (𝐹‘(1^st ‘(ℎ‘(𝑖 + 1)))))} ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < )) Fn (𝐵 × 𝐵)))
13	4, 12	mpbiri 258	1 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 {crab 3415 ∪ ciun 4967 ↦ cmpt 5201 × cxp 5652 ran crn 5655 ∘ ccom 5658 Fn wfn 6526 –onto→wfo 6529 ‘cfv 6531 (class class class)co 7405 ∈ cmpo 7407 1^st c1st 7986 2^nd c2nd 7987 ↑_m cmap 8840 infcinf 9453 1c1 11130 + caddc 11132 ℝ^*cxr 11268 < clt 11269 − cmin 11466 ℕcn 12240 ...cfz 13524 Basecbs 17228 distcds 17280 Σ_g cgsu 17454 ℝ^*_𝑠cxrs 17514 “_s cimas 17518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-inf 9455 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-fz 13525 df-struct 17166 df-slot 17201 df-ndx 17213 df-base 17229 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-imas 17522

This theorem is referenced by: imasf1oxmet 24314 imasf1omet 24315 xpsdsfn 24316

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