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Theorem imasdsval 17591

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)))))}

**Assertion**
Ref	Expression
imasdsval	⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 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) 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 imasdsval Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

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	1, 2, 3, 4, 5, 6	imasds 17589	. 2 ⊢ (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^, < )))
8		simplrl 789	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → 𝑥 = 𝑋)
9	8	eqeq2d 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘(1^st ‘(ℎ‘1))) = 𝑥 ↔ (𝐹‘(1^st ‘(ℎ‘1))) = 𝑋))
10		simplrr 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑌)
11	10	eqeq2d 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘(2^nd ‘(ℎ‘𝑛))) = 𝑦 ↔ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = 𝑌))
12	9, 11	3anbi12d 1465	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ 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))))) ↔ ((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)))))))
13	12	rabbidv 3425	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → {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)))))} = {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)))))})
14		imasdsval.s	. . . . . . 7 ⊢ 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)))))}
15	13, 14	eqtr4di 2818	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → {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)))))} = 𝑆)
16	15	mpteq1d 5203	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → (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))) = (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))))
17	16	rneqd 5930	. . . 4 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 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))) = ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))))
18	17	iuneq2dv 4983	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ∪ 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))) = ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))))
19	18	infeq1d 9445	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 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^, < ) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < ))
20		imasdsval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
21		imasdsval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
22		xrltso 13182	. . . 4 ⊢ < Or ℝ^*
23	22	infex 9462	. . 3 ⊢ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < ) ∈ V
24	23	a1i 11	. 2 ⊢ (𝜑 → inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < ) ∈ V)
25	7, 19, 20, 21, 24	ovmpod 7571	1 ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3081 {crab 3418 Vcvv 3457 ∪ ciun 4958 ↦ cmpt 5194 × cxp 5661 ran crn 5664 ∘ ccom 5667 –onto→wfo 6538 ‘cfv 6540 (class class class)co 7419 1^st c1st 7990 2^nd c2nd 7991 ↑_m cmap 8830 infcinf 9408 1c1 11116 + caddc 11118 ℝ^*cxr 11257 < clt 11258 − cmin 11456 ℕcn 12248 ...cfz 13551 Basecbs 17291 distcds 17341 Σ_g cgsu 17515 ℝ^*_𝑠cxrs 17576 “_s cimas 17580

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-fz 13552 df-struct 17229 df-slot 17264 df-ndx 17276 df-base 17292 df-plusg 17345 df-mulr 17346 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-imas 17584

This theorem is used by: imasdsval2 17592

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