imasdsval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > imasdsval		Structured version Visualization version GIF version

Theorem imasdsval 16621

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 16619	. 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 773	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → 𝑥 = 𝑋)
9	8	eqeq2d 2807	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘(1^st ‘(ℎ‘1))) = 𝑥 ↔ (𝐹‘(1^st ‘(ℎ‘1))) = 𝑋))
10		simplrr 774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑌)
11	10	eqeq2d 2807	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘(2^nd ‘(ℎ‘𝑛))) = 𝑦 ↔ (𝐹‘(2^nd ‘(ℎ‘𝑛))) = 𝑌))
12	9, 11	3anbi12d 1429	. . . . . . . 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 3428	. . . . . . 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	syl6eqr 2851	. . . . . 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 5056	. . . . 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 5697	. . . 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 4854	. . 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 8794	. 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 12388	. . . 4 ⊢ < Or ℝ^*
23	22	infex 8810	. . 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 7165	1 ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ^_𝑠 Σ_g (𝐸 ∘ 𝑔))), ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ∀wral 3107 {crab 3111 Vcvv 3440 ∪ ciun 4831 ↦ cmpt 5047 × cxp 5448 ran crn 5451 ∘ ccom 5454 –onto→wfo 6230 ‘cfv 6232 (class class class)co 7023 1^st c1st 7550 2^nd c2nd 7551 ↑_𝑚 cmap 8263 infcinf 8758 1c1 10391 + caddc 10393 ℝ^*cxr 10527 < clt 10528 − cmin 10723 ℕcn 11492 ...cfz 12746 Basecbs 16316 distcds 16407 Σ_g cgsu 16547 ℝ^*_𝑠cxrs 16606 “_s cimas 16610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-inf 8760 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-fz 12747 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-plusg 16411 df-mulr 16412 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ds 16420 df-imas 16614

This theorem is referenced by: imasdsval2 16622

W3C validator