bdmetval - Intuitionistic Logic Explorer

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Theorem bdmetval 14820

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 6072	. . 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 11448	. . 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 5934	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐶𝑦) = (𝐴𝐶𝐵))
9	8	preq1d 3706	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → {(𝑥𝐶𝑦), 𝑅} = {(𝐴𝐶𝐵), 𝑅})
10	9	infeq1d 7087	. . 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 6056	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ) ∈ ℝ^) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ^*, < ))
13	1, 2, 7, 12	syl3anc 1249	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 1364 ∈ wcel 2167 {cpr 3624 × cxp 4662 ⟶wf 5255 (class class class)co 5925 ∈ cmpo 5927 infcinf 7058 ℝ^*cxr 8077 < clt 8078

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-sup 7059 df-inf 7060 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-rp 9746 df-xneg 9864 df-seqfrec 10557 df-exp 10648 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181

This theorem is referenced by: bdbl 14823

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