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Theorem imasdsval2 17560

Description: The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised 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)
imasdsval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
imasdsval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
imasdsval.s	⊢ 𝑆 = {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)))))}
imasds.u	⊢ 𝑇 = (𝐸 ↾ (𝑉 × 𝑉))

**Assertion**
Ref	Expression
imasdsval2	⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔))), ℝ^, < ))

Distinct variable groups: 𝑔,ℎ,𝑖,𝑛,𝐹 𝑅,𝑔,ℎ,𝑖,𝑛 𝜑,𝑔,ℎ,𝑖,𝑛 ℎ,𝑋,𝑛 𝑆,𝑔 𝑔,𝑉,ℎ ℎ,𝑌,𝑛 Allowed substitution hints: 𝐵(𝑔,ℎ,𝑖,𝑛) 𝐷(𝑔,ℎ,𝑖,𝑛) 𝑆(ℎ,𝑖,𝑛) 𝑇(𝑔,ℎ,𝑖,𝑛) 𝑈(𝑔,ℎ,𝑖,𝑛) 𝐸(𝑔,ℎ,𝑖,𝑛) 𝑉(𝑖,𝑛) 𝑋(𝑔,𝑖) 𝑌(𝑔,𝑖) 𝑍(𝑔,ℎ,𝑖,𝑛)

**Proof of Theorem imasdsval2**
Step	Hyp	Ref	Expression
1		imasbas.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imasbas.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasbas.f	. . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
4		imasbas.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
5		imasds.e	. . 3 ⊢ 𝐸 = (dist‘𝑅)
6		imasds.d	. . 3 ⊢ 𝐷 = (dist‘𝑈)
7		imasdsval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		imasdsval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		imasdsval.s	. . 3 ⊢ 𝑆 = {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)))))}
10	1, 2, 3, 4, 5, 6, 7, 8, 9	imasdsval 17559	. 2 ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < ))
11		imasds.u	. . . . . . . . . 10 ⊢ 𝑇 = (𝐸 ↾ (𝑉 × 𝑉))
12	11	coeq1i 5836	. . . . . . . . 9 ⊢ (𝑇 ∘ 𝑔) = ((𝐸 ↾ (𝑉 × 𝑉)) ∘ 𝑔)
13	9	ssrab3 4038	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ((𝑉 × 𝑉) ↑_m (1...𝑛))
14	13	sseli 3935	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑆 → 𝑔 ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)))
15		elmapi 8834	. . . . . . . . . 10 ⊢ (𝑔 ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) → 𝑔:(1...𝑛)⟶(𝑉 × 𝑉))
16		frn 6703	. . . . . . . . . 10 ⊢ (𝑔:(1...𝑛)⟶(𝑉 × 𝑉) → ran 𝑔 ⊆ (𝑉 × 𝑉))
17		cores 6240	. . . . . . . . . 10 ⊢ (ran 𝑔 ⊆ (𝑉 × 𝑉) → ((𝐸 ↾ (𝑉 × 𝑉)) ∘ 𝑔) = (𝐸 ∘ 𝑔))
18	14, 15, 16, 17	4syl 20	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝑆 → ((𝐸 ↾ (𝑉 × 𝑉)) ∘ 𝑔) = (𝐸 ∘ 𝑔))
19	12, 18	eqtrid 2812	. . . . . . . 8 ⊢ (𝑔 ∈ 𝑆 → (𝑇 ∘ 𝑔) = (𝐸 ∘ 𝑔))
20	19	oveq2d 7416	. . . . . . 7 ⊢ (𝑔 ∈ 𝑆 → (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔)) = (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔)))
21	20	mpteq2ia 5200	. . . . . 6 ⊢ (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔))) = (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔)))
22	21	rneqi 5918	. . . . 5 ⊢ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔))) = ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔)))
23	22	a1i 11	. . . 4 ⊢ (𝑛 ∈ ℕ → ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔))) = ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))))
24	23	iuneq2i 4974	. . 3 ⊢ ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔))) = ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔)))
25	24	infeq1i 9427	. 2 ⊢ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔))), ℝ^, < ) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < )
26	10, 25	eqtr4di 2818	1 ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔))), ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∀wral 3079 {crab 3417 ⊆ wss 3907 ∪ ciun 4952 ↦ cmpt 5186 × cxp 5650 ran crn 5653 ↾ cres 5654 ∘ ccom 5656 ⟶wf 6521 –onto→wfo 6523 ‘cfv 6525 (class class class)co 7400 1^st c1st 7972 2^nd c2nd 7973 ↑_m cmap 8812 infcinf 9389 1c1 11089 + caddc 11091 ℝ^*cxr 11230 < clt 11231 − cmin 11429 ℕcn 12224 ...cfz 13526 Basecbs 17259 distcds 17309 Σ_g cgsu 17483 ℝ^*_𝑠cxrs 17544 “_s cimas 17548

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-inf 9391 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-fz 13527 df-struct 17197 df-slot 17232 df-ndx 17244 df-base 17260 df-plusg 17313 df-mulr 17314 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-imas 17552

This theorem is referenced by: imasdsf1olem 24491

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