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Theorem prdsdsval 16119

Description: Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Hypotheses**
Ref	Expression
prdsbasmpt.y	⊢ 𝑌 = (𝑆X_s𝑅)
prdsbasmpt.b	⊢ 𝐵 = (Base‘𝑌)
prdsbasmpt.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdsbasmpt.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
prdsbasmpt.r	⊢ (𝜑 → 𝑅 Fn 𝐼)
prdsplusgval.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
prdsplusgval.g	⊢ (𝜑 → 𝐺 ∈ 𝐵)
prdsdsval.d	⊢ 𝐷 = (dist‘𝑌)

**Assertion**
Ref	Expression
prdsdsval	⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ))

Distinct variable groups: 𝑥,𝐵 𝑥,𝐹 𝑥,𝐺 𝜑,𝑥 𝑥,𝐼 𝑥,𝑉 𝑥,𝑅 𝑥,𝑆 𝑥,𝑊 𝑥,𝑌 Allowed substitution hint: 𝐷(𝑥)

Proof of Theorem prdsdsval Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsbasmpt.y	. . 3 ⊢ 𝑌 = (𝑆X_s𝑅)
2		prdsbasmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
3		prdsbasmpt.r	. . . 4 ⊢ (𝜑 → 𝑅 Fn 𝐼)
4		prdsbasmpt.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
5		fnex 6466	. . . 4 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V)
6	3, 4, 5	syl2anc 692	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
7		prdsbasmpt.b	. . 3 ⊢ 𝐵 = (Base‘𝑌)
8		fndm 5978	. . . 4 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
9	3, 8	syl 17	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼)
10		prdsdsval.d	. . 3 ⊢ 𝐷 = (dist‘𝑌)
11	1, 2, 6, 7, 9, 10	prdsds 16105	. 2 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < )))
12		fveq1 6177	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
13		fveq1 6177	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
14	12, 13	oveqan12d 6654	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)) = ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥)))
15	14	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)) = ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥)))
16	15	mpteq2dv 4736	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))))
17	16	rneqd 5342	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) = ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))))
18	17	uneq1d 3758	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}) = (ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}))
19	18	supeq1d 8337	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^, < ) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^, < ))
20		prdsplusgval.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
21		prdsplusgval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
22		xrltso 11959	. . . 4 ⊢ < Or ℝ^*
23	22	supex 8354	. . 3 ⊢ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ) ∈ V
24	23	a1i 11	. 2 ⊢ (𝜑 → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ) ∈ V)
25	11, 19, 20, 21, 24	ovmpt2d 6773	1 ⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1481 ∈ wcel 1988 Vcvv 3195 ∪ cun 3565 {csn 4168 ↦ cmpt 4720 dom cdm 5104 ran crn 5105 Fn wfn 5871 ‘cfv 5876 (class class class)co 6635 supcsup 8331 0cc0 9921 ℝ^*cxr 10058 < clt 10059 Basecbs 15838 distcds 15931 X_scprds 16087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1720 ax-4 1735 ax-5 1837 ax-6 1886 ax-7 1933 ax-8 1990 ax-9 1997 ax-10 2017 ax-11 2032 ax-12 2045 ax-13 2244 ax-ext 2600 ax-rep 4762 ax-sep 4772 ax-nul 4780 ax-pow 4834 ax-pr 4897 ax-un 6934 ax-cnex 9977 ax-resscn 9978 ax-1cn 9979 ax-icn 9980 ax-addcl 9981 ax-addrcl 9982 ax-mulcl 9983 ax-mulrcl 9984 ax-mulcom 9985 ax-addass 9986 ax-mulass 9987 ax-distr 9988 ax-i2m1 9989 ax-1ne0 9990 ax-1rid 9991 ax-rnegex 9992 ax-rrecex 9993 ax-cnre 9994 ax-pre-lttri 9995 ax-pre-lttrn 9996 ax-pre-ltadd 9997 ax-pre-mulgt0 9998

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1484 df-ex 1703 df-nf 1708 df-sb 1879 df-eu 2472 df-mo 2473 df-clab 2607 df-cleq 2613 df-clel 2616 df-nfc 2751 df-ne 2792 df-nel 2895 df-ral 2914 df-rex 2915 df-reu 2916 df-rmo 2917 df-rab 2918 df-v 3197 df-sbc 3430 df-csb 3527 df-dif 3570 df-un 3572 df-in 3574 df-ss 3581 df-pss 3583 df-nul 3908 df-if 4078 df-pw 4151 df-sn 4169 df-pr 4171 df-tp 4173 df-op 4175 df-uni 4428 df-int 4467 df-iun 4513 df-br 4645 df-opab 4704 df-mpt 4721 df-tr 4744 df-id 5014 df-eprel 5019 df-po 5025 df-so 5026 df-fr 5063 df-we 5065 df-xp 5110 df-rel 5111 df-cnv 5112 df-co 5113 df-dm 5114 df-rn 5115 df-res 5116 df-ima 5117 df-pred 5668 df-ord 5714 df-on 5715 df-lim 5716 df-suc 5717 df-iota 5839 df-fun 5878 df-fn 5879 df-f 5880 df-f1 5881 df-fo 5882 df-f1o 5883 df-fv 5884 df-riota 6596 df-ov 6638 df-oprab 6639 df-mpt2 6640 df-om 7051 df-1st 7153 df-2nd 7154 df-wrecs 7392 df-recs 7453 df-rdg 7491 df-1o 7545 df-oadd 7549 df-er 7727 df-map 7844 df-ixp 7894 df-en 7941 df-dom 7942 df-sdom 7943 df-fin 7944 df-sup 8333 df-pnf 10061 df-mnf 10062 df-xr 10063 df-ltxr 10064 df-le 10065 df-sub 10253 df-neg 10254 df-nn 11006 df-2 11064 df-3 11065 df-4 11066 df-5 11067 df-6 11068 df-7 11069 df-8 11070 df-9 11071 df-n0 11278 df-z 11363 df-dec 11479 df-uz 11673 df-fz 12312 df-struct 15840 df-ndx 15841 df-slot 15842 df-base 15844 df-plusg 15935 df-mulr 15936 df-sca 15938 df-vsca 15939 df-ip 15940 df-tset 15941 df-ple 15942 df-ds 15945 df-hom 15947 df-cco 15948 df-prds 16089

This theorem is referenced by: prdsdsval2 16125 xpsdsval 22167

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