Theorem addsproplem6 28007

Description: Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.)

Hypotheses

Ref	Expression
addsproplem.1	⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
addspropord.2	⊢ (𝜑 → 𝑋 ∈ No )
addspropord.3	⊢ (𝜑 → 𝑌 ∈ No )
addspropord.4	⊢ (𝜑 → 𝑍 ∈ No )
addspropord.5	⊢ (𝜑 → 𝑌 <s 𝑍)
addsproplem6.6	⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))

Assertion

Ref	Expression
addsproplem6	⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem addsproplem6

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addspropord.3	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No )
2		addspropord.4	. . . 4 ⊢ (𝜑 → 𝑍 ∈ No )
3		addsproplem6.6	. . . 4 ⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))
4		addspropord.5	. . . 4 ⊢ (𝜑 → 𝑌 <s 𝑍)
5		nodense 27737	. . . 4 ⊢ (((𝑌 ∈ No ∧ 𝑍 ∈ No ) ∧ (( bday ‘𝑌) = ( bday ‘𝑍) ∧ 𝑌 <s 𝑍)) → ∃𝑚 ∈ No (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
6	1, 2, 3, 4, 5	syl22anc 839	. . 3 ⊢ (𝜑 → ∃𝑚 ∈ No (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
7		addsproplem.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
8		addspropord.2	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ No )
9	7, 8, 1	addsproplem3 28004	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})))
10	9	simp1d 1143	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ No )
11	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ No )
12	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
13	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑋 ∈ No )
14		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ No )
15		unidm 4157	. . . . . . 7 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑚))) = (( bday ‘𝑋) +no ( bday ‘𝑚))
16		simprr1 1222	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑚) ∈ ( bday ‘𝑌))
17		bdayelon 27821	. . . . . . . . . 10 ⊢ ( bday ‘𝑚) ∈ On
18		bdayelon 27821	. . . . . . . . . 10 ⊢ ( bday ‘𝑌) ∈ On
19		bdayelon 27821	. . . . . . . . . 10 ⊢ ( bday ‘𝑋) ∈ On
20		naddel2 8726	. . . . . . . . . 10 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
21	17, 18, 19, 20	mp3an 1463	. . . . . . . . 9 ⊢ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
22	16, 21	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
23		elun1 4182	. . . . . . . 8 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
24	22, 23	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
25	15, 24	eqeltrid 2845	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑚))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
26	12, 13, 14, 14, 25	addsproplem1 28002	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑚) ∈ No ∧ (𝑚 <s 𝑚 → (𝑚 +s 𝑋) <s (𝑚 +s 𝑋))))
27	26	simpld 494	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ No )
28		uncom 4158	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) = ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌)))
29	28	eleq2i 2833	. . . . . . . . . . 11 ⊢ (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) ↔ ((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))))
30	29	imbi1i 349	. . . . . . . . . 10 ⊢ ((((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
31	30	ralbii 3093	. . . . . . . . 9 ⊢ (∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
32	31	2ralbii 3128	. . . . . . . 8 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
33	7, 32	sylib 218	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
34	33, 8, 2	addsproplem3 28004	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑍)𝑔 = (𝑋 +s ℎ)})))
35	34	simp1d 1143	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ No )
36	35	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ No )
37	9	simp3d 1145	. . . . . 6 ⊢ (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
39		ovex 7464	. . . . . . 7 ⊢ (𝑋 +s 𝑌) ∈ V
40	39	snid 4662	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)}
41	40	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)})
42		oldbday 27939	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
43	18, 14, 42	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
44	16, 43	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( O ‘( bday ‘𝑌)))
45		simprr2 1223	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑌 <s 𝑚)
46		breq2 5147	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (𝑌 <s 𝑎 ↔ 𝑌 <s 𝑚))
47		rightval 27903	. . . . . . . . . 10 ⊢ ( R ‘𝑌) = {𝑎 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑌 <s 𝑎}
48	46, 47	elrab2 3695	. . . . . . . . 9 ⊢ (𝑚 ∈ ( R ‘𝑌) ↔ (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑌 <s 𝑚))
49	44, 45, 48	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( R ‘𝑌))
50		eqid 2737	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)
51		oveq2 7439	. . . . . . . . 9 ⊢ (ℎ = 𝑚 → (𝑋 +s ℎ) = (𝑋 +s 𝑚))
52	51	rspceeqv 3645	. . . . . . . 8 ⊢ ((𝑚 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
53	49, 50, 52	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
54		ovex 7464	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) ∈ V
55		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑔 = (𝑋 +s 𝑚) → (𝑔 = (𝑋 +s ℎ) ↔ (𝑋 +s 𝑚) = (𝑋 +s ℎ)))
56	55	rexbidv 3179	. . . . . . . 8 ⊢ (𝑔 = (𝑋 +s 𝑚) → (∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ) ↔ ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ)))
57	54, 56	elab 3679	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)} ↔ ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
58	53, 57	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})
59		elun2 4183	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)} → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
60	58, 59	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
61	38, 41, 60	ssltsepcd 27839	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑚))
62	33	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
63	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑍 ∈ No )
64	62, 13, 63	addsproplem3 28004	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑍)𝑔 = (𝑋 +s ℎ)})))
65	64	simp2d 1144	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)})
66	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑌) = ( bday ‘𝑍))
67	16, 66	eleqtrd 2843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑚) ∈ ( bday ‘𝑍))
68		bdayelon 27821	. . . . . . . . . . 11 ⊢ ( bday ‘𝑍) ∈ On
69		oldbday 27939	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑍) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑍)))
70	68, 14, 69	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑍)))
71	67, 70	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( O ‘( bday ‘𝑍)))
72		simprr3 1224	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 <s 𝑍)
73		breq1 5146	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (𝑎 <s 𝑍 ↔ 𝑚 <s 𝑍))
74		leftval 27902	. . . . . . . . . 10 ⊢ ( L ‘𝑍) = {𝑎 ∈ ( O ‘( bday ‘𝑍)) ∣ 𝑎 <s 𝑍}
75	73, 74	elrab2 3695	. . . . . . . . 9 ⊢ (𝑚 ∈ ( L ‘𝑍) ↔ (𝑚 ∈ ( O ‘( bday ‘𝑍)) ∧ 𝑚 <s 𝑍))
76	71, 72, 75	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( L ‘𝑍))
77		oveq2 7439	. . . . . . . . 9 ⊢ (𝑑 = 𝑚 → (𝑋 +s 𝑑) = (𝑋 +s 𝑚))
78	77	rspceeqv 3645	. . . . . . . 8 ⊢ ((𝑚 ∈ ( L ‘𝑍) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
79	76, 50, 78	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
80		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑐 = (𝑋 +s 𝑚) → (𝑐 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
81	80	rexbidv 3179	. . . . . . . 8 ⊢ (𝑐 = (𝑋 +s 𝑚) → (∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
82	54, 81	elab 3679	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
83	79, 82	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)})
84		elun2 4183	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
85	83, 84	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
86		ovex 7464	. . . . . . 7 ⊢ (𝑋 +s 𝑍) ∈ V
87	86	snid 4662	. . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
88	87	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
89	65, 85, 88	ssltsepcd 27839	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) <s (𝑋 +s 𝑍))
90	11, 27, 36, 61, 89	slttrd 27804	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
91	6, 90	rexlimddv 3161	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
92	1, 8	addscomd 28000	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
93	2, 8	addscomd 28000	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
94	91, 92, 93	3brtr4d 5175	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070   ∪ cun 3949  {csn 4626   class class class wbr 5143  Oncon0 6384  ‘cfv 6561  (class class class)co 7431   +no cnadd 8703   No csur 27684   <s cslt 27685   bday cbday 27686   <<s csslt 27825   O cold 27882   L cleft 27884   R cright 27885   +_s cadds 27992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec2 27982  df-adds 27993

This theorem is referenced by:  addsproplem7  28008