bdmetval - Intuitionistic Logic Explorer

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Theorem bdmetval 15168

Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)

**Hypothesis**
Ref	Expression
stdbdmet.1	⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ^*, < ))

**Assertion**
Ref	Expression
bdmetval	⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐶,𝑦 𝑥,𝐵,𝑦 𝑥,𝑅,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐷(𝑥,𝑦)

**Proof of Theorem bdmetval**
Step	Hyp	Ref	Expression
1		simprl 529	. 2 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
2		simprr 531	. 2 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
3		simpll 527	. . . 4 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐶:(𝑋 × 𝑋)⟶ℝ^)
4	3, 1, 2	fovcdmd 6149	. . 3 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐶𝐵) ∈ ℝ^)
5		simplr 528	. . 3 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝑅 ∈ ℝ^)
6		xrmincl 11772	. . 3 ⊢ (((𝐴𝐶𝐵) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ) ∈ ℝ^*)
7	4, 5, 6	syl2anc 411	. 2 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ) ∈ ℝ^*)
8		oveq12 6009	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐶𝑦) = (𝐴𝐶𝐵))
9	8	preq1d 3749	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → {(𝑥𝐶𝑦), 𝑅} = {(𝐴𝐶𝐵), 𝑅})
10	9	infeq1d 7175	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → inf({(𝑥𝐶𝑦), 𝑅}, ℝ^, < ) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ))
11		stdbdmet.1	. . 3 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ^*, < ))
12	10, 11	ovmpoga 6133	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ) ∈ ℝ^) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ^*, < ))
13	1, 2, 7, 12	syl3anc 1271	1 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 {cpr 3667 × cxp 4716 ⟶wf 5313 (class class class)co 6000 ∈ cmpo 6002 infcinf 7146 ℝ^*cxr 8176 < clt 8177

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 ax-arch 8114 ax-caucvg 8115

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-isom 5326 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-frec 6535 df-sup 7147 df-inf 7148 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-2 9165 df-3 9166 df-4 9167 df-n0 9366 df-z 9443 df-uz 9719 df-rp 9846 df-xneg 9964 df-seqfrec 10665 df-exp 10756 df-cj 11348 df-re 11349 df-im 11350 df-rsqrt 11504 df-abs 11505

This theorem is referenced by: bdbl 15171

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