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Theorem imasdsval2 17738

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⟶onto𝐵)
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⟶onto𝐵)
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 17737	. 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 5848	. . . . . . . . 9 ⊢ (𝑇 ∘ 𝑔) = ((𝐸 ↾ (𝑉 × 𝑉)) ∘ 𝑔)
13	9	ssrab3 4010	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ((𝑉 × 𝑉) ↑_m (1...𝑛))
14	13	sseli 3909	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑆 → 𝑔 ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)))
15		elmapi 8907	. . . . . . . . . 10 ⊢ (𝑔 ∈ ((𝑉 × 𝑉) ↑_m (1...𝑛)) → 𝑔:(1...𝑛)⟶(𝑉 × 𝑉))
16		frn 6747	. . . . . . . . . 10 ⊢ (𝑔:(1...𝑛)⟶(𝑉 × 𝑉) → ran 𝑔 ⊆ (𝑉 × 𝑉))
17		cores 6269	. . . . . . . . . 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 2810	. . . . . . . 8 ⊢ (𝑔 ∈ 𝑆 → (𝑇 ∘ 𝑔) = (𝐸 ∘ 𝑔))
20	19	oveq2d 7467	. . . . . . 7 ⊢ (𝑔 ∈ 𝑆 → (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔)) = (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔)))
21	20	mpteq2ia 5206	. . . . . 6 ⊢ (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔))) = (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔)))
22	21	rneqi 5931	. . . . 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 4978	. . 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 9508	. 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 2816	1 ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝑇 ∘ 𝑔))), ℝ^, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 {crab 3416 ⊆ wss 3900 ∪ ciun 4956 ↦ cmpt 5192 × cxp 5656 ran crn 5659 ↾ cres 5660 ∘ ccom 5662 ⟶wf 6562 ⟶ontowfo 6564 ‘cfv 6566 (class class class)co 7451 1^st c1st 8033 2^nd c2nd 8034 ↑_m cmap 8885 infcinf 9470 1c1 11240 + caddc 11242 ℝ^*cxr 11381 < clt 11382 − cmin 11580 ℕcn 12375 ...cfz 13679 Basecbs 17431 distcds 17481 Σ_g cgsu 17661 ℝ^*_𝑠cxrs 17722 “_s cimas 17726

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-map 8887 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-inf 9472 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-fz 13680 df-struct 17369 df-slot 17404 df-ndx 17416 df-base 17432 df-plusg 17486 df-mulr 17487 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-imas 17730

This theorem is used by: imasdsf1olem 24747

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