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Theorem imasdsval2 17144

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 17143	. 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 5757	. . . . . . . . 9 ⊢ (𝑇 ∘ 𝑔) = ((𝐸 ↾ (𝑉 × 𝑉)) ∘ 𝑔)
13	9	ssrab3 4011	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ((𝑉 × 𝑉) ↑_m (1...𝑛))
14	13	sseli 3913	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑆 → 𝑔 ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)))
15		elmapi 8595	. . . . . . . . . 10 ⊢ (𝑔 ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) → 𝑔:(1...𝑛)⟶(𝑉 × 𝑉))
16		frn 6591	. . . . . . . . . 10 ⊢ (𝑔:(1...𝑛)⟶(𝑉 × 𝑉) → ran 𝑔 ⊆ (𝑉 × 𝑉))
17		cores 6142	. . . . . . . . . 10 ⊢ (ran 𝑔 ⊆ (𝑉 × 𝑉) → ((𝐸 ↾ (𝑉 × 𝑉)) ∘ 𝑔) = (𝐸 ∘ 𝑔))
18	14, 15, 16, 17	4syl 19	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝑆 → ((𝐸 ↾ (𝑉 × 𝑉)) ∘ 𝑔) = (𝐸 ∘ 𝑔))
19	12, 18	eqtrid 2790	. . . . . . . 8 ⊢ (𝑔 ∈ 𝑆 → (𝑇 ∘ 𝑔) = (𝐸 ∘ 𝑔))
20	19	oveq2d 7271	. . . . . . 7 ⊢ (𝑔 ∈ 𝑆 → (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔)) = (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔)))
21	20	mpteq2ia 5173	. . . . . 6 ⊢ (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔))) = (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔)))
22	21	rneqi 5835	. . . . 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 4942	. . 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 9167	. 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 2797	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 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 {crab 3067 ⊆ wss 3883 ∪ ciun 4921 ↦ cmpt 5153 × cxp 5578 ran crn 5581 ↾ cres 5582 ∘ ccom 5584 ⟶wf 6414 –onto→wfo 6416 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 ↑_m cmap 8573 infcinf 9130 1c1 10803 + caddc 10805 ℝ^*cxr 10939 < clt 10940 − cmin 11135 ℕcn 11903 ...cfz 13168 Basecbs 16840 distcds 16897 Σ_g cgsu 17068 ℝ^*_𝑠cxrs 17128 “_s cimas 17132

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-imas 17136

This theorem is referenced by: imasdsf1olem 23434

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